While we have a gutshot, our specific combo is a prime overbet candidate on the turn to pressure the BB's condensed range.
Sizing up to an overbet is an aggressive deviation; while we have a range advantage, our specific hand prefers checking or a small sizing to keep the pot manageable with just a gutshot.
Note: Overbetting here is unnecessary; a small sizing or check better suits our hand's equity and the board texture.
Checking is fine, but this is a premier spot to overbet. We have the nut-straight blocker (A) and a gutshot, allowing us to polarize and fold out BB's weak pairs.
Once we check the turn and the board pairs, we have zero showdown value and must check-fold. Our hand has 0.4% equity against the range that reached this river.