Highlighting Thick Fat here because it is strictly better than Heatproof for what we want.
While Bulletproof can give us a full on immunnity to Focus Blast and Aura Sphere, and Filter can ease the damage somewhat of all super effective attacks, I want to showcase Thick Fat and Gluttony and speak about why these abilities should not be left in the dust.
While my comrade has shown us several calcs involving Thick Fat, I believe they have failed to highlight just what we can actually do with what Thick Fat provides.
As is all the cases of these secondary abilities, this one relies on a sort of surprise factor in order to be very successful, as realistically, on its own, Levitate outclasses every single ability currently up for discussion. Thick Fat can shine however, in punishing the opponent for assuming wrong. Now, before everyone comes up and start screaming uncompetitive!!!!!, Thick Fat will only be truly targeting a few key matchups, in a similar way in how Bulletproof targets a few matchups in exchange for Levitate's sweeping benefits, providing a worthy exchange.
These key match ups are Smokomodo and Heatran.
Thick Fat, more so than Bulletproof, relies on the success of Levitate to function to full capacity, and is specifically designed to lure in Pokémon like Smokomodo and Heatran. A Smokomodo that comes to revenge kill a CAP 26 has a choice in what move it can choose to use to eliminate it. Now, if what we predict is right, thanks to the ubiquious nature of Levitate and the massive risk of using Earthquake on a Ground type levitator, Smokomodo will near ALWAYS find that Flare Blitz will be the safer play against a CAP 26. This is where Thick Fat shines, as it does this in our matchup.
252 Atk Smokomodo Flare Blitz vs. 248 HP / 68 Def CAP: 350-414 (85.5 - 101.2%) -- 6.3% chance to OHKO after Leftovers recovery
vs
252 Atk Smokomodo Flare Blitz vs. 248 HP / 68 Def Thick Fat CAP: 174-206 (42.5 - 50.3%) -- guaranteed 3HKO after Leftovers recovery
Moving from a solid chance to OHKO after Rocks, to a 3HKO. What this gives the CAP 26 user is the chance to use a turn it would normally never have against Smokomodo and utilize it well - it can set a Doom Desire, using the frail Smokomodo and placing it in its own checkmate situation. Or it can use this one time chance and score an important KO that it would most likely never have the chance to do with Levitate. This is similar with the Heatran situation, in which does this
252+ SpA Heatran Magma Storm vs. 248 HP / 88 SpD CAP : 288-342 (70.4 - 83.6%) -- guaranteed 2HKO after trapping damage
252+ SpA Heatran Inferno Overdrive (180 BP) vs. 248 HP / 88 SpD CAP: 518-612 (126.6 - 149.6%) -- guaranteed OHKO
vs
252+ SpA Heatran Inferno Overdrive (180 BP) vs. 248 HP / 88 SpD Thick Fat CAP: 260-308 (63.5 - 75.3%) -- guaranteed 2HKO after Leftovers recovery
252+ SpA Heatran Magma Storm vs. 248 HP / 88 SpD Thick Fat CAP: 146-174 (35.6 - 42.5%) -- guaranteed 3HKO after Leftovers recovery and trapping damage
With Heatran being in the same pressure as Smokomodo, but even worse, as even if it predicts wrong, Earth Power will never OHKO a bulky CAP 26, placing even more pressure on it to use Fire type moves, giving us the oppurtunity to set Doom Desire and exploit Heatran's numerous common weaknesses, or just outright revenge kill it so we can be more at ease. Now these strategies are obviously not foolproof. Thanks to CAP 26 being slower, it can't abuse this relationship repeatedly, and it may outright fall if Levitate is revealed to be absent prior to the encounter. But I believe these two matchups, in addition to the general benefits of taking half damage from Fire and Ice type attacks provide us throughout the match, at least gives Thick Fat the strength to stand up amongst the other discussed abilities.
Now for Gluttony. Immediate recovery has been a hot debate for CAP 26, as some people believe that recovery moves may directly compete with and even force out Doom Desire on a moveslot. Now this is not the time to discuss the merits of said issue (we can do that later), it was brought up early on that if an ability can provide us with some form of recovery, we may help us pivot into more Pokémon and last longer, which is why Regenerator was brought up. Now, it was deemed that Regenerator competes too heavily with Levitate and was thus banned, I provide a softer, alternative method. Berry abusing abilities. Specifically, Gluttony, the one with the most potential.
Now with Gluttony, CAP 26 can have a form of pseudo recovery, triggering pinch berries at 50%, allowing CAP 26 to be overall last longer throughout the match, especially in combination with a certain move. Triggering at 50% health is huge for CAP 26's longevity, as it can be shown here.
If a Pokémon hits us with an attack that does 56% damage from full gealth, Gluttony triggers and we eat a pinch berry that leaves us at 94% health, practically making it as if we did not take any meaningful damage that turn, and allow us to retaliate appropriately.
Now if we did not have Gluttony, an attack that does say, 76% damage from full, would leave us with 74% maximun HP from full. Now I dont think I need to highlight how good this is for CAP 26, especially that since, in reality, it is much easier to work around a regular pinch berry by using attacks that do around 55 or 60% damage (which most strong attacks will be doing to us), effectively 2hkoing us through the berry. This also means Pinch berries can be used to heal through more than persistent chip and super effective blows that do around 70% damage, greater extending our matchup against more Pokémon, giving us more chances to set DD. Ofc, like most abilities being discussed here, it has signoficant drawbacks. It doesn't actually lessen the damage we take from moves like others here do, so very strong super effective attacks will always be killing us. It is crippled by the ever present Knock Off, limiting CAP 26's success against several Pokémon. Butt even with these flaws Gluttony can still stand toe to toe with abilities such as Bulletproof, Solid Rock and especially Sturdy. Like seriously, Sturdy?
Anyways thank for taking the time to read this, and I can't wait to have more discussions once moves are open!