Poison
I'm not really sure what made me go for Poison as the first round, especially since Zangoose is immune to getting poisoned. There's a range of different foes, though - some bulky, some fast - and I dislike that Gengar. Thankfully, this wasn't much trouble.
There are a range of good Normal moves to pick from - Secret Power is a useful move in buildings (it's basically just Body Slam); Double-Edge and Return give more power (but are more work); Rock Climb has an occasionally useful secondary effect (but has poor accuracy); Facade can come in very useful (but generally I'm not moving second and I'm immune to one status anyway). I split the difference and opted for Strength.
Due to Zangoose's general frailty, this was the only round I really felt comfortable running a Choice Band. For future rounds; I generally went with a Focus Sash.
Psychic
Again, Psychic has a very wide spread of foes and it's hard to cover them all with just four moves unless you've got an overwhelming level advantage. I considered running Counter just for Metagross, but this either required breeding or a trip back in time to Gen III and I eventually decided to just trust to being able to survive Meteor Mash+Bullet Punch. Predictably, I never fought a Metagross either here or in the Steel round. Never saw Bronzong, Latios, Latias, Cresselia, Jynx, Azelf, Uxie, or Mesprit - sometimes the Hall blesses you with easy foes. Espeon was the rank 10 fight, which has been tricky before but at this stage of the game was far outsped before it could start with any evasion nonsense.
Ground
Ground has a lot of really bulky species in it, so I wanted this out of the way. Astonishingly, Zangoose learns Giga Drain, though for all the Ground/Water species Strength actually outdamages it anyway. It was only really useful for Rhyperior, which otherwise possibly survives Ice Punch.
Steel
Steel is the first round where I ran a different EV spread, going for one which forwent speed in order for a generic boost to my bulk. The reason for this was largely that I was still slightly paranoid of Metagross getting a high roll with Meteor Mash+Bullet Punch. The increased bulk spread came in handy for a few later rounds too, though.
Don't think anything here was too much trouble in the end.
Water
As I've done before, I went with Water for the fifth round to avoid any crippling overspecialisation ruining my chances against Argenta. Expert Belt would have been a useful option to nudge some 2HKOs to OHKOs, but the uncertainty of what she'd bring pushed me towards caution. The only truly concerning foe is Poliwrath, whose Focus Blast+Vacuum Wave I'd be helpless to stop unless the former missed. Thankfully, no sign of it.
Ironically, she ended up using a Huntail. Still, not going to complain about that.
Rock
After floating the possibility of running a Liechi Berry - both for a one-shot nuke with Natural Gift and the possibility of a late-game Attack boost if required - I concluded Giga Drain would suffice. Most Rock-types are vulnerable on the Special side, after all. Most of the stuff I'd usually fear here when using other Pokemon (Aerodactyl, Tyranitar, Armaldo sometimes) aren't dreadfully problematic to Zangoose when it's holding a Focus Sash so this round passed without much incident. Never saw Shuckle or Regirock, which were the only foes I anticipated possible trouble from.
Normal
I truly dislike this round. Normal should be easy - basically everything is vulnerable to Fighting or Rock/Ice/Electric - but there's so much bulky stuff that can screw with you. Thankfully, Low Kick was given out to just about everything in Gen IV and that helps a lot. Snorlax is, for once, not a major threat - Staraptor stops Thunderpunch OHKOing thanks to Intimidate, but then helpfully uses Close Combat to make Quick Attack guaranteed to finish the job. I very nearly went with the bulkier EV spread before realising at the last minute that otherwise my Quick Attack might not hit it first. Thankfully, my instincts were right as Staraptor ended up being my rank 10 fight.
Fighting
You might have been wondering why I left Zangoose's sole weakness as long as this. The simple reason is that - as I said in my notes - it's not actually that complex of a round.
That said, there's a couple of problematic foes here. Poliwrath, as before, is a pain; so is Machamp. Wrestling with how to handle this one gave me cause to think I probably had, after all, left Fighting too late. But even at the lowest possible level, Aerial Ace comes nowhere near an OHKO - and then you've got to take a Dynamicpunch to the face. Use a Chople Berry or a Focus Sash, and you're still confused. Only a Choice Banded Giga Impact can hope to take it out - but running a Choice Band is risky when there's so many other foes to contend with for whom other items are a better choice. I even considered running a Jolly variant to outspeed Infernape, but you just need the extra power Adamant provides.
In the end, my best hope was that if I encountered a Machamp it'd have Guts and miss its first Dynamicpunch. Ultimately, Machamp never appeared. But I do think if I was doing this a second time I'd opt to shift Fighting further back. On balance Poison certainly doesn't need to be done first.
Fire
Infernape is the only truly threatening foe here. It's simple enough to face down - Protect into Fake Out, take the hit from Close Combat, then KO with Dig - so this could have been done even later. But you don't want to leave some of these too late - Houndoom speed ties at level 25, but any later and you run the risk of getting critted or burned. Entei is no joke with that annoying Quick Claw, too. This is another round that makes going Jolly might be helpful but, again, you want as much power as you can squeeze.
Electric
Which brings me neatly on to Electric. Decided to go for the bulky spread for this round - so many Electric-types are too fast to have a hope of outspeeding, so there's really no point investing in Speed at all here. Doing this meant I had a much better shot at surviving Zapdos' Thunder and - crucially - avoiding full paralysis. Which is exactly what happened when I faced Zapdos as my rank 9 fight. Jolteon was the rank 10 and had my gnawing my fingernails when it got off 2 Double Teams and avoided my first Dig, then hit me with Thunder Wave - thankfully my second Dig hit.
This might be my least favourite round full-stop, no matter what I use.
Flying
Whereas I think I often overestimate this one. With a balanced spread of moves that hit everything super-effectively and the reassurance that Focus Sash means I'll be able to tank most hits, it's not too tricky. Salamence looks fearsome but even after Intimidate Ice Punch+Quick Attack is sufficient to take it down; Togekiss is also a rather tricky foe, but most of the other stuff here is easily managed.
Ice
Another round it may have served me better to do a little earlier. While Weavile isn't as threatening as it is to a lot of Pokemon and doesn't do enough damage to bait itself into using Ice Shard, a lot of Blizzards are very powerful so I opted to go bulky here. Carrying a Lum Berry over a Focus Sash made me uneasy, but a random freeze would have ruined everything. It also guards against Jynx's Lovely Kiss.
Zangoose's main problem is that it gets so many mid-powered moves like Brick Break and the elemental punches, and so few over 90 BP. The high defense stats many Ice-types share mean that Rock Slide, Fire Punch, and Brick Break didn't always look like sure-fire answers. Of course, Zangoose gets Close Combat - but only at a high level, and as stated I prefer to play the Hall at as low a level as possible. You can breed for it, of course, but on balance I felt that Close Combat's defense-lowering side effect was an unnecessary risk given I'm not carrying a Focus Sash here and it only offers a tangible advantage against Regice, who is otherwise manageable. Not worth the time investment ultimately.
Grass
Decided to opt for Grass next due to there being a few bulky mons I didn't want to face right at the end of the challenge. Since a lot of foes are either too slow to outspeed (Venusaur, Tangrowth, Meganium, Exeggutor, Victreebel, Torterra) or too vulnerable for their speed advantage to pose an issue (Breloom, Roserade, Jumpluff, Sceptile) I considered going with the bulky spread but ultimately decided against it as my immunity to poison meant that errant status inflictions from moves like Sludge Bomb weren't a concern. It might have been more prudent to go for bulk ultimately, though. Meganium was the only truly concerning foe, but chose Energy Ball rather than going for Counter - though even if it had, I'd still have won.
Dark
I initially wrote off Dark as a round I wouldn't have too much trouble with and left it late. After all, nearly everything here has exploitable weaknesses and is outsped. However, looking back at the list of foes, I realised I'd miscalculated when it came to Spiritomb.
Here the problem of mid-powered moves hits me again. Very little Zangoose can muster can hope to do more than 33% to it - but its Dark Pulse does a tremendous amount of damage. Even with a bulky spread, it has the possibility of doing over 50% with a single hit, so there's the chance it'll bring me down before I can 3HKO with X-Scissor.
After some thought, I decided to run a Sitrus Berry instead of the usual Focus Sash. On balance definitely a good choice as this meant I was in far less danger of losing to Houndoom if it used Dark Pulse and got a flinch - even burned, Brick Break and Quick Attack are sufficient to 2HKO. After initially considering running Secret Power to avoid making contact with Sharpedo, I did the calculations and realised that Quick Attack could 2HKO anyway so there was no danger of losing to a random flinch. This also made it possible to 2HKO Skuntank, whom Secret Power could only 3HKO. Tyranitar's Stone Edge was still threatening, but with a Focus Sash I die anyway thanks to sandstorm damage. So that only left Crawdaunt, of all things, as the major threat - an easy kill in the Water round, but a 2HKO here thanks to that inconvenient defense investment. Crabhammer does a disarmingly high amount of damage, and if it scores a critical hit I'm finished.
Ironic, then, that I faced Crawdaunt as my rank 10 fight. Thankfully, Crabhammer did not crit.
Bug
Going bulky here was an option, but it doesn't really save me from anything that'd kill me anyway - Armaldo's X-Scissor and Stone Edge still 2HKO in either case, and Stone Edge probably just crits regardless. And it's generally safer to outspeed what you can, though Scyther outspeeds no matter what. Pinsir and Heracross are both lessons in how not to use a pinch berry: Pinsir would have been an interesting fight, but I never saw it.
Ghost
Given my general dislike of that Gengar, this would have been a candidate to do earlier. But it's so overspecialised that I felt I could handle it later. Ultimately I was correct, though more thanks to luck than considered planning.
Dusknoir forces you to run the bulkier spread - Brick Break does a phenomenal amount of damage, while Shadow Claw hovers between 2HKOing and 3HKOing based on the damage roll. With a bit of bulk, you open up the possibility of Brick Break doing less than 50% to you and falling short of a 3HKO. Basically it becomes a coin toss. So concerned was I about this that I actually contemplated going for full Attack/Defence investment before reconsidering - sure it brings the damage range down a little, but special bulk is useful in this round too.
Here, the "mid-powered moves" issue once again rears its head. While Shadow Claw is sufficient for the likes of Banette, it just doesn't quite do the trick elsewhere - Froslass survives it easily, and even Gengar, should it appear at level 30, has a chance to survive it. There's always Night Slash, which is another move Zangoose gets through breeding. But while it does boast a high critical-hit ratio, that's not something you want to depend on and its base power is identical to Shadow Claw so they just end up doing the same amount of damage to everything.
Then the answer hit me - Payback. Gengar and Froslass are going to outspeed you anyway, so why not? Carry a Lum Berry to prevent Blizzard from freezing and stop Hypnosis inflicting sleep - perfect.
And of course I got Dusknoir for my rank 10 fight... and it obligingly killed itself for me with Curse after my first Shadow Claw did just over 50%.
Dragon
Dragon was left for last because Ice Punch takes care of all but one of the species you'll encounter here. This left me free to make the rest of my moveset as broad as I could for Argenta, though really it was a question of what the fourth move should be since priority and a STAB attack seemed logical. Fighting had the best overall coverage with Ice so Brick Break was chosen.
After making it through the Dragon round with no issues (Kingdra dying in one hit to a crit Strength) I was ready to take on Argenta.
Who used... Lucario.
Well, that's not so bad. Lucario has no speed EVs here, so I comfortably outspeed it... mmm, Brick Break doesn't KO, though. That's less of an issue when you have Quick Attack to mop up the last few HP points, but with a low damage roll they'll do just under 100%.
And that's precisely what happened. Brick Break left Lucario on just under 20% - it struck me with Aura Sphere in return, only missing the OHKO thanks to my Focus Sash. Extremespeed doesn't yet have its heightened priority bracket so my Quick Attack went first, leaving it on what seemed like precisely 1 HP and causing me to get knocked out.
Ah, if only I'd taken the time to breed for Close Combat after all...
I can't even really be mad about this. Considering how many other Pokemon I've struggled with in the Hall, to get to 169 on a single attempt with Zangoose is absolutely nuts. Genuinely torn between trying again for 170 and just leaving it - the fact that I only lost on a technicality is bruising, but my overall goal is to get a streak of 10k and it's a matter of one single extra battle (plus what'll probably amount to two or three after).
