Considering the way the hand played, I would fold, expecting to see a 3, 5, or small pocket pair (77 or lower) from BB here. The line chosen forces us to rely solely on showdown value (we don't really have any) against a range that we either chop or lose to. I think the mistake on this hand comes on the turn, as the check in position indicates you don't have a T (capping your range and allowing Villian to bet the hands listed earlier confidently on river) or you are worried about a T, allowing Villian to introduce possible bluffs on the river. Would you ever check trips (even weak ones like T7s) on the turn in position with just straight draws present? On the turn, we can recognize this hand will probably finish as a bluff vs. the board. Since our flop action has set us on the bluffing path, why not continue?
I believe a more profitable line could look like this: Bet $0.70-$0.75 on the turn. Villian responds in following manner: Villian folds (Hero wins), raises (Hero folds), calls turn and checks river (Hero checks behind for showdown value), or calls turn and bets river (Hero folds). With this bet, Hero wins a larger pot those times Villian calls with overcards for two streets, actualizes any potential fold equity against a weak A or pocket pairs (99-66, 44, 22), and can represent top of range hands (quads, boats, and top trips down to T9, and pocket pairs), which would put Villian in a difficult spot if he is a thinking player. The line chosen doesn't afford us any of these opportunities and we have to rely solely on showdown value and we only beat K-high bluffs. While we do lose this hand occasionally, I believe we would end up on top because of the number of times Villian folds to the top card pairing on the turn.
Therefore, I think I would bet this hand on the turn in position, and then check this river card if we get to it.