I'd likely not CB this flop, which sucks - but we don't get a lot of hands to fold. If I did, I understand db'ing. On this river, it's a shit spot. I'd expect you to ship it with like all of your range so if you throw out like a couple $ bet, I'd expect him to ship a ton of his range (better+bluffs) & only call with worse. So IMO I think we should be checking the flop & betting small on this river. Any line IMO on the river is fine. If we c/f that's fine. If we 3bar AI, I don't see him folding a lot here which is why IMO that's probably our worst option.

I'd just hate checking for this reason: If he has no heart, he's likely shoving now. If he has a heart, he's still likely shoving now. So a huge % of his range is going to ship this river when checked to. That's why I'd much rather b/f ~3$ here. Yes our odds sucks because we'd likely want to call when shipped on, but it at least accomplishes something. When they call, we're likely ahead. When they ship, they are way ahead or way behind.

This spot really sucks though so I'm not really sure of our BEST option as all of them now suck.

It's unlikely that anyone would fold a flush for a 2/3 pot shove given the line/runout. If they were drawing w/ something like TxTh on the turn, that says a lot about how they view the turn situation and how they're likely to play rivers.

If he calls both flop and turn and unless he is compelete fish the most off his Jx without any hearths should fold on the turn because he is either crushed now(All Ax) or half the deck crushes him on the river. My estimate off his range on the river how the hand played out are following: 

All gutters that has one hearth any 48JA with a hearth and also complete flushes. 2pairs and sets SHOULD raise the turn so you wont get a cheap river. If he is tight you can actually bluff him off the lower flushes because your line looks very strong in my opinion. Like overs with a hearth or full boat rest is pure bluffs. That if he can think so far beucase the river is worse for you then for him and if u bet,bet,bet with that kind off board you are shoving much more strenght then weakness. Its not fun for him to call with a Th flush when your range is very polarized. 

But then again he might look at his flush think that you MIGHT be bluffing here or just have a lower one and call anyway. Against a good thinking player this is a good 3rd barell since it wides your range and also makes him fold lets say half off his flushes. Against a bad one its not working since he played the hand just like some kind draw with a pair and the river completes that for him.

I'd lean towards not betting this flop. As played, turn seems fine and I'd just ch the river and play pokers

Yeah agree with mplecki here betting the flop out of pos with 6 outs(you dont wanna see Ah,Qh anyway) is not as profitable as just c/f. He folds to c-bet about half the time but it is a 3-bet pot + he has pos AND you dont block anything useful.