Much tougher decision than in the first one, at least in my opinion. BB check could mean two things after his call:
a) He is slowplaying a monster and will C/R us
b) He is on a draw that has missed and doesn't want to invest a BB into the pot to see the river /indicate strength
Since we have shown strength by raising on the flop, and that the 4d is likely to be a blank for BB, I bet, because of two things:
a) Might get a better hand to fold (such as a higher flush draw or a pair)
b) I'm probably ahead with my draw or the BB thinks I'm ahead
Checking is also viable, but since our hand hasn't improved on the turn (not even our drawing odds
have improved), we get to see a free card. But checking here means that any cards that comes a blank will get the pot away from us (since it screams missed draw).
If opponent C/R, I go into the think tank and see if I have any read on him acting this way on the BB before.