You really should try raising at some point in the hand, otherwise you're playing in the dark. Check-calling OOP for so many streets is very stationish and will not yield any profit to you.

I think this is a fold. He is not going to be 3 barreling worse than AKxx here very often, unless he has a QJTx+fd(s) and is bluffing river. I think check raising flop or potting turn as played is much better. At worse he has KKs on the flop and you get it in with only flush draw equity and against AAs you are still behind but not by very much... obviously that becomes different after the turn and it would be pretty bad to get it in against either of those hands. I think check raising flop is probably best because of that. As Kusi said it is really hard to know where you are when you are check calling oop. I think #betting this pre is fine too, it will certainly define your opponent's hand better imo.