While our gutshot has equity, we must check our range on low, connected boards that favor the Big Blind's calling range.
Checking is the pure strategy here. This low, connected texture is much better for the BB's range than ours, as they have all the 78s, 23s, and 45s/56s combos.
Note: C-betting into a board that heavily favors the defender's range is a mistake; we should check to realize equity with our gutshot.
After betting the flop, the turn 4h pairs the board and we should continue our aggression with a larger sizing to maximize fold equity with our gutshot.
Checking is correct. The board has double-paired, and while we have two pair, any 6, 5, or 4 beats us, as do the straights that were already possible.
Folding is the only play. Our two pair is essentially a bluff-catcher that loses to almost every value hand the BB would bet for this size.