I'd think if the BB had an ace he would have 3-bet preflop to your semi-steal attempt or check/raised the flop.

I bet the turn again to charge gutshots and flush draws, with the intent to check behind on the river. If you're check/raised on the turn I think you can call with the # of outs you've got.

My concern with checking the turn is that many of your outs put a 4-straight or 3 flush to the board. I'd hate to take a free card that just helps someone else more than me given I might be ahead at the turn anyway.