This CAP concept is quite interesting to me. I like it a lot, but I have no idea how it would be done, so I'm just gonna kinda write a lot and see if that helps at all.
So, our concept is:
A Pokemon that dissuades your opponent from fainting it, or can even leave it's presence on the field felt even after it faints.
It's much easier to talk about the part after the comma, so I'll start with that.
Between status, hazards, weather, or hole-punching, it's easy to have the effects of a Pokemon felt. Deo-S was popular for dual screens, being almost guaranteed barring priority, just like with hazards. Deo-D was the same. Even after they fainted, their effects could be felt, given the fact that another member of their team could set up much easier, or gain crucial OHKOs.
In a slightly different vein, a parashuffler could wreak havoc on a team. If it's successful, you could have at least half your team paralyzed, and if you don't have a member to remove status, you're really hurting. When that parashuffler faints, you're really feeling it.
The thing with these examples is, you
want to KO these Pokemon as fast as possible, to stop them from doing more damage to your team. To me, the crucial part of the concept is to make your opponent to
not want this CAP to faint. It's a simple idea, but quite difficult in practice. Aftermath is one thing that may dissuade your opponent from fainting you, but it's only 25% of your health, and only on a contact move. Against more than half of Pokemon, Aftermath is useless. On those it does affect, that 25% probably won't save you. Being 6-5 with one Pokemon at, say, 50% is way better than still being at 6-6.
Moves such as Grudge or Destiny Bond (I see that we should avoid discussion, but I have to mention them too. Sorry) also do this, but they take a turn to do, plus only last for one attack. They are also very niche, and creating a Pokemon around these moves would be poor indeed. We want this Pokemon to useful while alive, not just spamming one move repeatedly. This CAP isn't about making the match 5-5.
So how do we make this Pokemon fainting advantageous to the user? I have no idea, short of creating a new move or ability, but I have no idea what the protocol for that is, so that's all I'll say about that.
Somebody earlier in the thread mentioned that a fainted Pokemon is a pretty big deal: you get a free switch-in to try to stop whatever just fainted your last Pokemon. That's why revenge killers are a thing. Sometimes, you purposely faint a Pokemon to safely bring in another to set up and try to end the game there! That's a big deal. Memento and Healing Wish are great in assisting a set up for a sweep. Memento to bring in a Pokemon to set up on their otherwise counter, forcing the opponent to switch twice to let that previously Mementoed Pokemon do its job. Healing Wish fully restores a sweeper, allowing it to set up while fully healthy. While useful, and certainly in line with part two of our concept, it doesn't make your opponent not want to KO it; quite the opposite! It wants to KO that Pokemon so it can't KO itself.
tl;dr: I've said everything everyone else has said. I tried.
What is CAP 19 doing while it's not being KO'd?
I think this CAP would be best suited to be a support Pokemon. Your opponent would really have to ask if it's worth KOing it, if it's better dead or alive to you, and to them.