Stacks:
UTG - UTG ($7.84)
MP - MP ($6.08)
CO - CO ($5.29)
BTN - BTN ($31.43)
SB - SB ($10.70)
BB - Hero ($7.59)

Preflop: ($0.07, 6 players) Hero is BB with Qs Qd
3 folds, BTN raises to $0.15, 1 fold, Hero raises to $0.47, BTN calls $0.32

Flop: 4s 8d 5s ($0.96, 2 players - Hero: $7.12, BTN: $30.96)
Hero bets $0.70, BTN calls $0.70

Turn: Ah ($2.36, 2 players - Hero: $6.42, BTN: $30.26)
Hero bets $2.00, BTN calls $2.00

River: 5c ($6.36, 2 players - Hero: $4.42, BTN: $28.26)
Hero bets $3.05, BTN calls $3.05

Total Pot: $12.46
Hero shows Qs Qd (two pair, Queens and Fives)
BTN shows As 2s (two pair, Aces and Fives)

BTN wins $11.94

Villains stats: VPIP 43, PFR 8, Hands 170, FCB 19, 3B 2, F3B 0, BSA 27. BTN PFR 14.

My question simply is: should I have shoved rather than bet with the sizing I put in? Given how he was calling the flop with most of his range anyways, I felt his range was pretty weak. He has a few sets in his range, but no over pairs that beat me. Other than that he's on flush draws. Thanks :)