KK , river ?

I would check on turn and b/f on the river

on the turn BB checks, this is a sign that he has some kind of draw, so I bet to charge draws and protect my hand

Hi.

Turn is a clear check. You bet only when you have value from a calling range. Don't just assume that people have draws just because they check. He would probably check a set here on the Turn as well when the turn brought a 4th straight card.

Your decision should be based on whether you can get called by worse hands more than 50% of the time.

Given the 3way dynamic, I'll just check/fold on the Turn. BTN can still have straights and flushes, I don't think they have too many worse hands, and their weaker pocket pairs are probably folding the Turn if you bet.

So we cannot bet for value, our hand is too strong to bet as a bluff, so we check.
When someone bets, we don't beat enough bluffs, and playability is really poor on the river. So we check and fold as opposed to check/call.