You're in position on this flop against a coordinated hand in his top 10%. More importantly though he cbets the flop half the time, so if it had completely missed him, he is probably check/folding or check/calling this flop.
We have 12 outs to the nuts, and 0 showdown equity and are 200 bets deep in position. This is a great spot for a float on the flop, if the flop was rainbow I would say go ahead and raise this flop, but with the 2 hearts it allows our opponent to play back at us with a wider range of hands. (AAxx no flush draw might not be happy about shoving this flop, but with a flush draw it's an insta shove).
We're in position, let's go ahead and use this position to call the flop and reconsider on the turn. If I thought opponent had a flush draw, I would also be flatting this flop with 2 pair, maybe even a set (for balance reasons) as well as draws like this so that opponents have to barrel again into us on the turn, which when 200 bets deep isn't a good proposition.
Anytime you find yourself out of position with aces, and flop a flush draw 200 bets deep, it's awesome to be able to shove this flop, as you often aren't doing that badly (only losing to a set). We should be looking to force our opponent to playing out of position on multiple streets given his cbet on the flop makes up half of his 3 betting range so I expect him to have a piece of it (he doesn't just blindly cbet OOP)
On this board I believe I am calling the flop bet and calling the turn bet and probably giving up on the river since I don't expect a bluff to ever work here (Villain probably check/calls this river to induce bluffs off busted draws, since our opponents will percieve our range is never 2 pair or a set on this flop after flatting generally, so the 4 pairing isn't scary at all).