C-bet flop with BDFD and gutshot, double barrel turn with turned double gutshot and heart draw, and barrel river.
Check-raising flop is an option, though it sucks if we get reraised and can't realize our backdoor equity. So is folding as you played on the flop with your check. Calling is the worst of the four options. C-bet/barrel good turns > Check-raise > Fold > Calling. I'd prefer the check-raise more if we did not have any backdoor equity and if the A were instead a K, as there are way many more Ax in the BB's calling range than Kx. It is hard to make a pair though let alone top pair. And your hand will be disguised if you hit your gutshot. A lot of players will also check behind if you check-raise a dry board and check the turn, giving you a free river. I don't expect people to ever fold TP, so if I get called, I'm just giving up unless we improve.
All you have is a gutshot; you're not getting the right odds
to call. It's not like you're both super deep and he has a monster here often. He'll probably fold all his one pair hands other than Ax, and there are no draws he can call with.
Anyone finding a river fold if we 3-barreled this board run-out and got jammed on? Our range is uncapped, and we can easily have monsters like sets and backdoored flushes. We can't really beat any hands that raise the river for value, and river is a pretty bad spot for Villain to bluff at after we three-barrel such a runout. Bluff-catchers would just call down since we're polarized to nuts/air, people don't really turn hands with SDV into bluffs yet, there aren't really any straight/flush draws out there, and I don't think the BB would call with small suited connectors pre.