QQ vs super nit; Line check

Im out of position, in the SB. CO, who is 4 and 4 over 106, opens. I put him on AQ+, TT+. I don't like the way i played the hand. I think i should have 3bet PF, which prob. chases out the BB and gives me info on the villains hand; a call and im good, a shove and im fairly certain he has KK AA and i fold. Flatting like i did allows the BB to come along and i have no idea where i stand. Again, on the flop i should have bet for info. Be bets and im facing 2.8:1. If i put him on AK, TT+ i have 38% equity, if i expand that to AK, TT+, 88, 55 i have 33%. Hope im doing this right; shoving is the same as if he had shoved and i called? 3.35/7.6=.44? So i needed >44 equity to shove here?

Full Tilt No-Limit Hold'em, $0.10 BB (8 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

Button ($9.70)
Hero (SB) ($9.95)
BB ($3.90)
UTG ($10)
UTG+1 ($10.65)
MP1 ($14.25)
MP2 ($10.15)
CO ($3.65)

Preflop: Hero is SB with Q, Q
4 folds, CO bets $0.30, 1 fold, Hero calls $0.25, BB calls $0.20

Flop: ($0.90) K, 5, 8 (3 players)
Hero checks, BB checks, CO bets $0.50, Hero raises to $4, 1 fold, CO calls $2.85 (All-In)