This is the exact spot I struggle with too in the SB spot and have a good hand to start with. I recently had a post and the 3bet Pre is proper but that isn't where you were at, you flat the call from btn which put you in a wierd spot for the flop.

Now….The btn raise (a.k.a. his cbet) was air for the BB exposed that with his lil raise. What would your result be if you 3 bet his raise instead of BB? If you had re raised BTN, you would would have had a better read of the BB because at this point, in the preflop you wouldn't have much read to know where you are at. This is why you need to try and lead out Pre as best as you can with your AJ so you can get your reads out before the river. It IS possible that BB has nothing and the BTN as resulted that BB beat you to the read of the BTN (he had nothing).

Now you are faced with another “investment” in chips to see where are at to see if the BB was simply still had air which could have been a three bet air dung bet to take it down before the next turn with absolutely nothing or he hit the flop and crushed. 

At the spot and where you are at in the hand. Your Pre 3 bet would have given you a more desirable result of the hand. Since you flat the BTN, and played as is. I would simply fold in your spot and look for another spot.