River bet --- straight --- paired board

[holdin2]

Villian 17/0, over 230 hands, wins 88% at showdown
Was this weak? The check/call, check/call, lead made me think "let's see this to showdown, I don't want to raise and have to call a reraise".

Ultimate Bet No-Limit Hold'em, $.25 BB (7 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

MP2 ($27.75)
Hero ($23.55)
Button ($13.20)
SB ($55.30)
BB ($29.60)
UTG ($17.75)
MP1 ($9.75)

Preflop: Hero is CO with A, J.
2 folds, Hero raises to $0.85, 2 folds, BB calls $0.60.

Flop: ($1.80) K, Q, Q (3 players)
BB checks, Hero bets $1, BB calls $1.

Turn: ($3.80) T (3 players)
BB checks, Hero bets $2.5, BB calls $2.50.

River: ($8.80) 7 (3 players)
BB bets $2, Hero calls $2.

Final Pot: $12.80

[MikeJee]

You played this hand perfectly except for the river IMO, why didn't you raise this? Were you scared of a full house? (obviously) I can't fathom just flat calling him here. What went through your mind? Hit me.

[freechus9]

I like the flat call on the river. What would be your plan if checked to instead?

[holdin2]

You played this hand perfectly except for the river IMO, why didn't you raise this? Were you scared of a full house? (obviously) I can't fathom just flat calling him here. What went through your mind? Hit me.

Here's what went through my mind. I am about done for the night, I've had a good session and don't want to ruin it. There is a paired board, the villian doesn't raise or reraise at all preflop, so villian could easily have KK, KQ, Q10s, 1010. As I said, the check/call, check/call lead made me wary.

Now as I haven't totally jacked up the pot, they could be playing AA, AK, or any queen (or AJ) and think they own me and want to get some value by leading the river, but I'd rather get to showdown than get in it for stacks.

I like the flat call on the river. What would be your plan if checked to instead?

I'd probably value bet about half pot. Any significant raise would put me just about all in, but I would think trips or a full house would want to lead the river. Suggestions and reasoning?