Big post supporting Thick Fat
I don't think that we should give Thick Fat to the CAP and here are the reasons:
1. Thick Fat doesn't allow the CAP to deal with anything that it wants to and can't already beat. All of your examples of special attackers that carry Ice moves, such as Politoed, Thundurus-T, and Starmie, are already walled. While Thick Fat will make it easier to tank Ice Beam and HP Ice, without huge passive recovery from Sitrus Berry or insta-heal and status immunity from LumRest, the CAP will be easier to deafeat in the long run, as its most viable form of recovery will most likely be Morning Sun when using Thick Fat(i don't think that anyone wants to give Recover to the CAP right?). So overall, the CAP performs better against the special attackers that it wants to beat with Harvest than with Thick Fat. Now let's talk about the other effect of Thick Fat, the Fire damage halving. Why would we want to give to the CAP this privilege? Is the CAP supposed to tank any Fire move, except from Latios's HP Fire in sun, which can already be handled by the CAP? No. So why allow it to do so? Why give it a trait that makes it better in every kind of team except for sun teams? So, in the end, Thick Fat doesn't give anything that helps the CAP fulfill its role that Harvest doesn't already.
2. Thick Fat makes the CAP much more viable outside of Sun, while not helping sun at all, as i already explained in my first post. With Thick Fat, the CAP now has only 3 relevant weaknesses (i am excluding Poison for obvious reasons), and 5 very relevant and useful resistances/immunities (Water, Ground, Grass, Electric, Psychic), as well as immense special bulk, and possibly the ability to rapid spin and beat every Ghost in OU, and Pursuit trap. Why do you want to give to the CAP an ability that only enhances its performance outside of sun, but does nothing to help the CAP with the threats that we want it to (when compared to Harvest of 'course), and nothing to help sun teams in general? One of our goals is exactly the opposite of what Thick Fat does, and this goal is to make the CAP boost sun's usage and performance more than it will boost the same attributres of other types of teams, such as sand and rain. This is why we want abilities that are mostly useful inside sun.
Honestly, I'm not sure I'm entirely convinced by any of these abilities. The niche benefits of Leaf Guard, Infiltrator, Sticky Hold, Synchronize, etc. sound cool and all, but is that really worth burning off this ability slot? Is it worth making the artists deal with something they might not be prepared for, just for some very specific niche? I mean, sure, art is a distant second priority in a competitive discussion, but I'm bringing it up to demonstrate just how pointless I think most of these abilities are.
If we have to pick a competitive secondary ability, then Sticky Hold and Synchronize seem like the best bets to me. Sticky Hold exists mainly to block Trick, which is used on about 22% of all Latios, and which would cripple a Harvest variant of CAP 5. Synchronize is a weird one, but I'm mainly interested in it because of Tentacruel. While a sun team does tend to have Will-O-Wisp and/or Lava Plume, having something that could cripple Tentacruel by switching into its Scald outside of sun could be somewhat beneficial to a sun team. Then again, it could also be beneficial to a sand team... I don't think that Infiltrator is really worth it, and Leaf Guard just seems like an excuse to use non-berry items, which I'm not really keen on making an a priori goal of the secondary ability.
Leaf Guard's benefits may be niche, but this is what we want from our secondary ability anyway right? Didn't we want either an ability of minor significance that will only help the CAP slightly with its role and won't overshadow Harvest? If yes is the answer, then Leaf Guard fits like a glove. It helps sets that want to use Leftovers for the constant passive healing (Sitrus doesn't always activate) by allowing them to prevent status when in sun, which helps against many of the Pokemon that we want to beat, such as defensive Waters, which commonly use Scald and Toxic. It also helps offensive sets with either LO or a choice item, but i don't think that those sets will be good, so i don't really care. It is situational sure, as sun won't always be up, but it does have some use, and it does help the CAP with its role, even if it helps just by a little. And let's not forget that many Ninetales use Sunny Day to keep their weather up as Politoed comes in, so Leaf Guard can actually be active sometimes when swithcing into Politoed. Oh and the CAP can use Sunny Day itself, to keep momentum for its team, as well as greatly benefiting itself from the sun, activating Leaf Guard and enhancing its recovery(Morning Sun). Finally, Leaf Guard is one of the few abilities of minor importance that we can give to the CAP that only works in sun, which is obviously something good.
So, while other minor abilities such as Sticky Hold and Synchronize do help the CAP a bit with its role, they help it in all enviroments, not just sun, and imo are even more situational than Leaf Guard.