KK on wet board OOP

As the pre-flop aggressor, I would CBet such a flop.
Then when you check the turn, opponent may check behind

You could also bet the turn.
Terrible river, considering it gives you a set

Bet/fold?
I'm stuck at that one

Theory wise we should not get raised on this flop almost ever as our range has huge advantage over sets and AK. That said I would imagine pool to easily think two pair or maybe even top pair is nuts and raise them often. Still we probably want to cbet as there are ton of weaker that will call. As played I like your call line on flop and turn.

River we improve absolute hand strength wise but hand relative strength doesn't change much. If we donk river it's tough to get called by weaker. So check and hope villain check. His bet range should be polarised into straight or bluff so our hand is bluff catcher. So I would not mind folding to sizable bet.

Reasonable alternative line is to use block bet on river like 1/4-1/3 pot to try getting value from two pairs with plan to fold to raise. This could also be done with Ax hand for same reason and for inducing bluff raise if we care about range protection.