Villain is solid, so I think we can assume he's re-popping the preflop with AQ+ and TT+. So our odds
going into the turn look like so:
Against AQ+ with 1 heart in their hand, we're ahead, and they have a 25% shot at beating us on the turn.
Against AQ+ with no hearts, they've got a 6% chance of beating us by the turn.
Against QQ+ with no hearts, we have a 21% draw to beat them on the turn.
Rest we either crush or get crushed by.
There's a large variation in our equity depending on villain's holding. Its going to be very difficult to get max value out of all of villain's range with 1 action.
Since we're OOP, in a 3-bet pot, with a decent draw that our opponent may not have, I think checking here gives us the most information & puts us in the best position. Lots of hands with flush draws are likely to raise us, and we're in a bad position to call a raise.
Additionally, villain's choice of betting or checking behind on this flop will give us some indication of if he has a flush draw or not. Would you check behind on this flop with [AcAs] and risk getting out-drawn? I doubt it. Thus, I think if we get a check behind us, we can narrow villain's range down a bit, and think about discounting our outs to the flush.