Overs + 2nd NFD on paired board

[Silly String]

Villain was unremarkable over only a dozen or so hands. Loose player who doesn't buy in full. Looking back his stats were 46/13.

I feel I played the flop right, but I am unsure about my turn play here. What's standard given the paired flop and villain's smooth call?

PokerStars No-Limit Hold'em, $1.00 BB (5 handed) Poker Stars Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

MP ($99.50)
Hero ($105.70)
SB ($69.25)
BB ($112.80)
UTG ($101.55)

Preflop: Hero is Button with K, J.
2 folds, Hero raises to $4, SB calls $3.50, 1 fold.

Flop: ($9) T, T, 4 (2 players)
SB bets $3, Hero raises to $12, SB calls $9.

Turn: ($33) Q (2 players)
SB checks, Hero ????

[Warpe]

If we raised to get a free card then let's take it. Any bet he calls here potfucks him so we have to be ready to call a shove.

You did notice you picked up the OESD here, right?

[Deanglow]

Take the free card and overflush him on the river.

[Silly String]

Yeah Warpe, picking up the OESD is what cause me to go into auto-pilot and bet the turn like Jo-Jo the idiot circus boy. I was reviewing the hand and thought that might have been a very stupid bet. That is why I posted it here to make sure this is a check behind despite the 15 outer.
My thinking goes like this: Against this loose opponent, I have to check behind, but against a very tight opponent I would probably fire a 2nd barrel. Sound as a pound?

[Warpe]

His flop bet and call screams a weak T to me.

[Irisheyes]

ya he's trying to trap you, don't let him.

[silu73]

Take the free card. Flop play is fine.

[mixchange]

I agree on the free card, if you bet you could get raised to a price that you realllly don't want to pay. I like flop play...check turn and hopefully take down a monster on river