I'm not sure, but I think we're losing value if we're not cbetting this flop.
On the flop my line would probably to bet/call without the Kh. I think villain could still just flat you pre with AA, although I wonder how often he is just flatting the 3bet from the cutoff with AA.
On the turn then I guess it's a fold if he then bets again expecting him to actually somehow have AA or TT.
I agree with you that we're probably not getting 3 streets of value in a 4bet pot with KK. bet/bet/check seems fine. I don't think people will bluff a lot when you bet flop bet turn, since they're repping super thin.
A lot of guessing as you can see
Not used to play 500bb's deep on 25NL
EDIT: therefore interested in others opinions obv