Check again. Just defend pre, this hand is a lot weaker than a raggedly ds kings would be in four card imo. You really want to get those five cards working together hands.
I think turn check is ok. Betting 1/3 to 1/2 psb with the intention of shutting down on river / folding to xr isn't a bad alternative.
Villain range is fairly queen heavy, but we have a pretty good bluff catcher on lots of rivers after checkback and can improve. The 7 is also nice.
Flop we can consider alternative sizings or checkback.
Agree this is not a great 3! Pre, too many low cards and third heart mean you are very unhappy facing a 4! and postflop playability is poor when villain calls.
I like pre - building a pot with DS KK in position is what we want to do. I think we have to bet the flop - our hand is very strong. Turn seems like an obvious check.