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I think you can 3bet this vs bu. His range is awfully weak and then you can easily take it down on the flop with a cbet (if he even calls).
As played you underrepresented your hand preflop. So your perceived range is on the lower side, which hits 57T quite well. Checking the flop is fine, but yes, I'd certainly lead out on the turn. The 2 doesn't change anything. He's more likely to have air or a T or a flushdraw on this flop and with a T or flushdraw he should cbet. You can represent a middling pocket pair, a T or a set as well as flushdraws. This flop is too wet for him to be checking behind with much of a hand. I wouldn't bet too much money on that he is doing that correctly, but I think we can definitely fire the turn and sometimes fire the river.