I'll just get right to the meat of this issue. Should you raise the turn? call the turn? or fold the turn?
I would call down as you did and expect to lose and heres why. Basically the only hands you are ahead of are hands that the BB could possibly overplayed on that flop. Hands like: KdQd,KsQs,AdJd,AhJh,AdTd,AhTh.

The hands that make more sense given the BB's flop play are AA,KK,QQ,AK,AQ and you have zero to 1 out against all these hands on the turn. Even if you assumed that the BB would play all the overplayed hands the same as the logical hands you are still a 13-6 underdog. All you have here is a crying call on the turn and river. Your one glimmer of hope is that the BB was overplaying his hand cuz he knew he probably had the all-in guy beat. This is a decent possibilty, but it doesnt change your call down strategy to a raise situation, all it does is give you a chance, the odds are still against you. But the pot is large enough to call down.