I struggle with these spots too. I wonder if AQ should really be in our turn bluff catching range since it in fact doesn't even beat all his bluffs. A better idea versus this opponent could be to add more strong hands in our flatting range pre-flop, so flatting AK/KK/AA and just start calling down with those instead.

U eather have to widen ur 4-bet range or tighten up in late position. If u play back at him he will realise this. AQ is not in bad shape against his 3-betting range. On the flop u really just got 2 options raise or fold. Calling souldn't be an option there. Calling  down with marginal hand normally doesnt beat rags!

Yeah ur right we give up position, but just calling will be -EV against such a rag we also have to use 4-bet bluffs i would say otherwise he keeps betting and we keep pay him of and with a hand as strong as AQ we cant just call call fold all the time. Tricky spot !

Preflop: even though i like calling the 3-bet. If one constructs GTO ranges its difficult not to use AQ as a 4-bet here when the BTN opens 50%+ and SB is 3-betting 18% - our 4-bet needs more value combos.

Postflop: As played i think its either a flop fold OR call flop fold turn depending on cbet freq. on this texture / barreling frequencies of villain. The main reasoning is: i'm assuming villain has his widest/most bluffs in these positions, which means he will cbet a J63r board all day, ALSO he will continue his bluffs on A/Q turns which are great for us. So we have built in implied odds when he cbets and gives up or continues barreling on favourable turns. Now, as Hero i can't see how a made hand would call the flop and fold the turn on a 2 AND the only real draw gets there if you called preflop with 54s. low pairs and SCs should be in your range. So his turn bet is really only folding out your pure floats OR he has something. The turn didn't bring a flush draw so villain has less semi-bluffs too. Respect the turn barrel and FOLD. 

someone said "Fold flop vs his barrel stats. You basically want to hit a pair and not
fold vs him. If you call flop with A high he is just going to make you
fold on a later street." - I think this is a good idea, what do u think?

I think calling this flop is fine. Even standard. 

I'm not a fan of a flop raise, in fact I think it's reasonable to not even have a raising range on this flop. A raise doesn't really rep anything on a board this dry. NOTE: not necessarily saying that a raise is definitely -ev or that not having a raising range is best - I just think calling plays better with most of our range.

I think a fold would be very weak on this flop, we know he has a reasonably wide range preflop and he is likely cbetting this flop a ton with his air. We rep a J well when we flat so he likely check folds a lot of undercard turns with his air (unless he picks up a draw). The cards he most likely barrels on with air are the A turn, the K and the Q. Which we are happy about on 2 of the 3. I would likely float again on the K turn given he barrels all his air and we pick up a nut gutter.

However I do think we should be folding this turn, it is the definition of a card he won't be barreling a lot with air, he can't even have picked up a backdoor flush draw.