I would probably start betting smallish with this combo. I'd rather ch/call with AQ or AK.
As played with his sizing he represents Jx+, but you can never know for sure against a fish.
The problem with calling is, that when he checks back twice he reps showdownable hand, and IMO it's kinda rare that someone bets this turn bluffheavily after checking back twice.
On the other hand you checked all 3 streets and he might attack this.

So all in all I would fold AT on the river since you'll have better combos to call with and you might even play for ch raise with some flushes.

I think CBing is fine as well as C/Cing. If CB, I don't really think you have too much fold equity until you get to river. So if we do CB you should be prepared to make multi barrel's and use board texture in your favor to now when to continue aggression. Villain will be over-pair saturated, and all his BDFD's will call with over cards to bout.... Most his weaker pairs are nuts, and 33 is a OESD.....
On turn we pick up equity , aaayyy!
and on river our hand blocks NF's & a bunch of villain wheels

wp

Yeah well played imo!

wp wp

This seems to be a common theme in uNL hands - this another board on which we should have a super-high cb frequency against a BTN r/c range. As played bet the turn big. Given we have a flop checking range, presumably it contains some AA/AKdd which now want to bet big; and so do nearly all of our bluffs.