I'd start with a flop raise. If you're even planning on value raising this spot, the big pairs are the best hands to do it with.

If you flat the flop I think you can also raise turn for value. KK should still be ahead of whatever range villain is using here.

Getting to the river this way, I think checking back is reasonable. You should both be afraid of the straight and that means villain might not call down super light. Villain could also be potentially going for a x/r with a 6x or better so I don't think that this river is always safe when checked to.

IMO i think this is fine. I think his river check signals 'medium strength hand' because all bluffs and nutted hands continue, so i would be tempted by a small value bet to get calls from all lower overpairs.

As for line. I like getting your trapping hat on. I think things that call a flop raise we can't bet against on the turn or river (obvs run out dependent) but it shows that keeping bruffs in V's range works well, sir.

Villain shouldn't have many 2x in his range and if he has it then I'd expect a bigger donk bet , so raising for value from smaller overpairs is good. No need to let him set a small bet size when he'll put more money in if you push him anyway.

As played no need to try and bet the river. He's going to fold his smaller overpairs and of course we're risking walking into a check raise some of the time.