As long as you're sometimes checking with hands that contain aces etc I check here (which I personally do)

But yea there is a point in betting, it certainly stops you from being bluffed off the best hand almost every time (every time he decides to bluff that is, not every time you check) and he folds any hand that isn't a set or contain an ace =/ how much of his range do you think that is? His button 3 bet of 15% and squeeze of 4-5% doesn't help a super large amount unless you know what kind of hands he's been 3 betting the button with, and also how often he 3 bets utg raises... Is he someone who squuzees button with loads of aces so he can pot control A high boards etc and block AK/AA etc or is he someone that will squeeze hands like 77 here etc. How often does he call 4 bets ?

So I think it boils down to if you have a decent play for the hand after you check you should check... if you have no idea wtf to do just bet something small.. I'd be interested to hear others opinions..

I've won a ton with KK on A high boards check calling 2 (sometimes 3) streets in 3 bet pots tbh, but its generally where I know alot about the player and know how I plan to proceed with the hand on certain turns etc... This here is a pretty tough spot with the small amount of information given imo

i dont like the size of your 4bet.You give him great odds and are oop if he calls.
I cbet here with almost any 2,inlcuding my 4bet bluff range.