In my opinion definitely fine to just flat the flop check-raise. Would anyone consider checking back the turn to induce some river bluffs?

The reason why I would always shove over his raise is that I really don't expect a villain like this to raise/fold hands that he would shove OTT once we call.
So if my assumptions are right we don't lose anything by shoving, but if we just call, a heart OTT can make him fold worse flushes and sets which we would have stacked.

With these said, against unknowns or from a non exploitative viewpoint we will have to just call X% of our nut flushes so that he can't just raise any2 and shove every blank turn.

As played I would ch back turn sometimes. By shoving you likely make him fold all his sets -if he has any- which isn't necessarily a bad thing(it's actually good if he doesn't make mistakes on rivers with them), but I think it is a bit more likely that he'll call worse flush if we shove OTR.

SPR on the turn would be less than 1:1 so villain shouldnt be x/raise/fold here unless he was bluffing and he doesnt bluff enough here to go for slowplays.

Think shoving flop makes more sense at this SPR than if we are deeper but don't hate calling.

On turn he likely checks Sets and his weaker flushes and maybe some bluffs though I don't think he has a ton of bluffs here. Think checking turn induces more bluffs and thinner calls than turn shove so I would check.

On turn he likely checks Sets and his weaker flushes and maybe some bluffs though I don't think he has a ton of bluffs here. Think checking turn induces more bluffs and thinner calls than turn shove so I would check.

The problem I see with checking back turn and betting river is that all his sets have at least 23% equity against us (9 combos), but I can only think of 4 combos that he'd play this way with which he'd be willing to pay off another bet OTR. KhTh (1 combo) and KhQx (3 combos). And it is questionable if he calls with those. I mean what bluffs are we supposed to have here after bet-calling flop, checking back turn, and shoving river?
I don't know but I can't think of any hands but the nuts.

By betting the turn we allow villain to make a mistake on the turn with his sets and worse. He needs 28% equity to call but his sets only have 23% against our hand. I would want to put villain in spots where he can make the biggest mistake possible. If we check back turn and jam the river he will have an easy time folding his sets and say "thanks for the free card". And we don't know if he's going to call Kh at any point after we called his flop x/r so his whole continuing range at this point might be just those 9 combos of sets. And if we check back the nuts here every time it is very bad for our EV against this range I believe.

I would shove the flop. As played turn is a weird spot. I would probably xb here because he's going to fold all his sets, 2pairs and he might calldown lighter OTR when we jam.