Given the fact he just calls your preflop raise (he has not every single Ax combos in his flatting range, I guess), and given the fact he rarely has a flush (he has more small flushes like 87hh than A5hh in his range, and he is almost always betting a small flush on the turn I think), I would call here. Otherwise your flop bet seems really senseless. It doesn't make much sense to bluff pure air on this wet board.

From a theoretical point of view we have to call I think, but it also depends on any reads u may have on villain (like timing tells, history, sizing tells and so on).

Like you said, complete is an option. I think it would be the best play, because villain does not fold and is passive (not gonna raise often when we complete).

Flop and turn well played, river is tough. We can fold imo. We should have a ton of Ax-hands in our range, and we'd like to have a heart blocker when we bluffcatch.

I'd 3 barrel this.
As played on the turn b/f river.
Or in your river case which is the worse line I can think of you should probably x/c with your bluffcatcher... I mean the whole hand doesn't make very much sense to me the way you played it tbh.

wp imho

cbet on the turn. check-call on the river.

WP

Absolutely never XC OTR. We need reads on villain to do that. And even than to easy for villain to have A2o here.

We should not bet OTR but I think we will often split or be beat, because all pair and 2-pair combos villain will bet OTT when checked to.

would checkfold flop, then bet bet if it checks, once he calls flop your kinda screwed oop