I definitely think there's at least on street of value in here. More if villain is aggro pre and will 4bet AK a majority of the time. Having 2 blockers to KQs really eliminated a lot of the hands that will be ahead of you on this runout.

Flop check seems good. If villain is good he can punish you for betting the turn by checkraising frequently. The more straightforward villain is the more I like a bet on the turn. Either way I would bet river for value. The lack of strong Kx in his range combined with his passive play that indicates he rarely has more than 1 pair and the number of JJ-88 combos he could have make it a pretty attractive bet.

Checking flop is standard.

On the turn as we don't expect MP to be calling weaker Kx pre and also 4 betting AK some % value betting twice looks good. We also shouldn't allow somebody to freeroll a heart on us.

The river only needs to be a small bet targeting his pocket pairs.

I would have bet the turn so as not to give a freebee runout or a chance to hit some heart draw and if I got raised fold and move on.
Not expecting a chk/r on the river so I'd bet small, say 60-80c and see if I could get a crying call from and underpair.

I agree with the above analysis. Flop is pretty much a mandatory check, turn should definitely be a bet IMO as we are not expecting to get raised here very often and we want to charge any hand he contains with a heart. He can have a wide range of underpairs that include a heart here that are almost certainly going to call at least one street.

River I'm betting again then, we can credibly represent a missed flush draw ourselves so it's very plausible that villain tries to pick us of with a weaker pocket pair here. Some people above have suggested a small sizing on the river, I don't think we want to make it that small though. Somewhere in the region of 50-60% of pot seems good to me as it's unlikely we'd go super small if we did miss a flush draw and were looking to bluff villain off a hand.