At the very least, I am back up to 10 wins, and have started recording enemy move sets in Word.
Already mapped by others in this thread. Collected in the first post. (It's the links before the actual records.) Sorry, if I realized you weren't using them, I would have already mentioned them.
Edit: So I watched the battle video you posted. There are two major errors I noticed.
First was the second turn Skill Swap from Audino after mega evolving. That was just wrong all around - with a confused teammate on the field, I feel like Heal Bell was the obvious call, but Heal Pulse would have also helped if you were hoping Healer would activate. I feel like that's what caused you to lose Slaking. Since Audino goes after Slaking, Heal Bell wouldn't have prevented that first self-hit, but at the very least it would have given you the chance to Slack Off + Heal Pulse on turn 2, which would have survived the double Payback and opened you up for Return + Heal Pulse on turn 3 (anticipating a Confuse Ray from Umbreon) which puts you in a relatively good position against this team: opening gambit successful, one pokemon up, and only one team member slightly injured. (Potentially confused, but that can be worked around.)
The second big mistake was trying to attack with a confused Aegislash who had the option to swap out. If Aegislash is confused, the
only correct moves for it are King's Shield and swapping out - confusion is
devastating to Blade Forme Aegislash, as you witnessed firsthand there.
In fact, in general, the correct answer to confusion in the Battle Maison is to swap out. Sometimes swapping can be dicey, but in many situations it is less dicey than staying in and hoping you don't take the confuse hit.
Also, Payback is a contact move, and Aegislash could have used King's Shield straight away to get an attack drop on Absol (and potentially Umbreon) so that the hit that activated Weakness Policy would have hurt even less. King's Shield + Heal Bell is a
really good combo if you get an attack drop out of it. I don't really know if that would have changed the outcome of your fight, but it's another option you could have taken to think about in the future.
I also notice Slaking didn't get Lefotvers recovery, and it seems to have dealt more to Absol1 than the math says it should have gotten. Did you put a Silk Scarf on it? If that's the case, a Life Orb would have been much better - you'd have had an 87% chance to score the OHKO instead of a mere 35%, and the OHKO probably would have saved this battle for you.
There was definitely some bad luck here (you took
every confusion hit possible), but there were also some misplays. Assuming that fourth mon is Victreebel1, this was probably winnable for you.
The good news is, your theorymoning for this team seems to have been pretty solid. I think Cleric Audino + First-turn-setup Mon + Aegislash is a good team core for doubles, you just need to find a setup sweeper and a fourth member who fit well together. (I would look at Gyarados for the setup sweeper first, just because Skill Swapping Intimidate is a gimmick I could not pass up, but I'm weird like that.)
Edit2: So, probably should have looked this up, but apparently Heal Bell does not cure Confusion. I still maintain that Heal Pulse was a much better action than Skill Swap, and double-switching your front line wouldn't have been terrible either.