$5 NLHE Full Ring: 3B AQo from SB vs UTG open and call and 4Bet from BB

Now I am okay with the 3bet in this situation but the 4Bet from BB got me in a corner. With the other 2 players folding there was some dead money in the pot still I was only getting ~37% odds to get BB all in. I thought of shipping lightly here bc BB was half stacked but in post analysis think it was a loose play. We might be slight underdog against up till +JJ and huge underdog to anything higher and AK. Also BB has been playing tight so far, don't know how he got half stacked but that's definitely not a good sign. Either way did we ship too lightly even for <60bb?

888 Poker - $0.05 NL - Holdem - 8 players
Hand converted by PokerTracker 4: http://www.pokertracker.com

UTG: 96.6 BB (VPIP: 28.51, PFR: 19.30, 3Bet Preflop: 5.13, hands: 228)
UTG+1: 65.4 BB (VPIP: 40.00, PFR: 6.67, 3Bet Preflop: 0.00, Hands: 16)
MP: 205.4 BB (VPIP: 36.54, PFR: 29.81, 3Bet Preflop: 15.91, Hands: 106)
MP+1: 103.6 BB (VPIP: 12.90, PFR: 8.06, 3Bet Preflop: 0.00, Hands: 64)
CO: 118.8 BB (VPIP: 12.96, PFR: 5.56, 3Bet Preflop: 4.55, Hands: 54)
BTN: 111.2 BB (VPIP: 19.35, PFR: 6.45, 3Bet Preflop: 0.00, Hands: 248)
Hero (SB): 173 BB
BB: 57.6 BB (VPIP: 16.06, PFR: 11.52, 3Bet Preflop: 3.64, Hands: 194)

Hero posts SB 0.4 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has Qc Ad
UTG raises to 2.8 BB, fold, MP calls 2.8 BB, fold, fold, fold, Hero raises to 12.2 BB, BB raises to 34.6 BB, fold, fold, Hero raises to 90.6 BB, BB calls 23 BB and is all-in

Meh, it's fine vs a shortstack. You'll be flipping a lot of the time, though.