[playtitleist]

I put villain in on this flop.

First, I can't put villain on trip Ks because he would have pushed pre-flop. Therefore, best for villain is trip 9s or 5s - not very likely based on pure chance (but so be it if that what they have, fire another one up). If you are betting into his trips, he raises all-in anyway, and you have to call. So might as well push him here, because you HAVE to bet, and here's why:

You need to make villain decide if they want to put their tourney life on the line right here. You can't give infinite odds to a player facing elimination. You are killing yourself if he has a PP that hits trips on the turn, or if his flush or gut-shot hits.

Bet it all-in. With villain's stack size, you are forced to call any turn/river bet, so only chance is to push the flop and hope villain folds or calls with something like AK.