I think you should raise the BBs open about 50% of the time and just flat the other 50% of the time. I think you played the hand fine. I'm never folding here when BB 4bets allin because I think he can also have overpairs here.
Our A
J
has 25% equity against a set
Our A
J
has 36% equity against Aces
Our A
J
has 46% equity against Kings or Queens
Based on math, we need 33% equity to call the all in. If V ever makes this play with a flush draw, Tens, or hand like A9, this is definitely a call.
That said, I don't think calling can ever be bad. This is a high variance spot so don't get upset when we lose occasionally.
Also, BB should have definitely bet more on the flop, atleast $25.