I think mix in checks is good but I also think in this spot you want to block villain's suited broadway combos with hearts because they have a lot of equity vs your hand. I haven't solved for this but I ran 40% open * (1-55%) for ~18% range to include all pockets, all suited aces, suited broadway, ATo, KJo+. I have your stack as 9.16 remaining and the pot odds as 9.16 / (9.16 + 4.23 + 9.16) = 40.6%.
Continues are AhXo, A9, K9, J9, 76, 77,66, all hands w/ 2 hearts or 2 diamonds, JTs. I think villains natural bluffs are AhXx that is not AhKo (AhTo+), KdTd, QdJd, QdTd, JTs (no hearts), 77, 66 maybe sometimes but they would probably just take showdown. He has all AXh except probably AKhh at less frequency which 4bets pre, all suited hearts broadways, A3s, maybe 76s, 43s at a low frequency (I gave him 50% combos of 76s). Sticking with your read that it's not a boat you're 61/39 vs that range but as soon as you take out diamond bluffs, JTs non-hearts out and he starts checking down his AhXo, it's narrowly -EV (38/62 vs range). The more he checks down ace high without the flush with the more disastrous calling is. If he's ch/b Axdd but bluffing non-A high XXdd pure it's a clear call (52/48 vs range). My read is population underbluffs so I probably cry and fold or cry and call in the moment. I am more inclined to call this vs a rec who is capable of showing up with 22, ripping A9 or A8 for no good reason than a reg, etc. Usually recs are not playing with a full stack and so you are getting better odds at this point is another small consideration.

The heart blocker is significant and also the higher your PP the better your blocker effect vs his 3b call and the more this is a call. (e.g. AhA has 57% vs {ch/dn A highs, jam non A-high diamonds, 50% 77, 66, 76, flush} and AA no heart only has 45%!} ThTx vs the same range has 28% equity. TT no heart has 18% equity!