A lot depends on if you believe the player is just calling you to play in position against you because he believes he can win a lot of hands by bluffing or he just happens to have two hands he wanted to play in this spot. Without any knowledge of the player for the first hand i'm checking since our hand has downgraded a little bit due to the flush draw and the over card on the flop. If he raises flop i call/fold depending on the size he bets. If you know you're going to fold to a bet on the turn when that heart comes I don't believe you should be thinking to bet unless you have fold equity and are trying to get him to fold meaning you should've bet bigger, or just have checked and see what he does.
You can consider checking the flop in these spots as well, to keep villains range wider. It may become trickier to play on later streets tho, potentially exposing you to aggression.