On the flop it is a matter of pot control. This board is very wet, and hits the blinds range way more than it hits yours. You have an over pair, which may still be good, but with too much action you will be forced to fold with the flop texture. Two hearts on board, you have the

for a backdoor draw. Flat call the flop.
I would imagine that if you flat the flop, the SB will probably check the turn.

hits your range, and SB also has to worry that you or BB called the flop with a heart draw, since as it turns out that he had no heart in his hand this is even more likely. BB, who also as it turned out had no heart, would also check the turn leaving the action on you with the decision to semi
bluff bet or check behind and try to bluff catch while also realizing your equity if you are behind.
At this point, the pot would have been 4,059 with effective stacks of 11,486. I would lean more toward a bet here as the

is a fantastic card for your range. You can represent AK with a

, or a hand like


or a hand like you have, big pair with a heart. The bet doesn't have to be big here, I would bet 30-50% pot, 1200-2000, leaning toward the bigger size if you want to be able to have a pot size all in bet for the river