Bet this rivered 2 pair?

BB is 23/15 over 50 hands, 7% 3bet, have barely seen him play out of the blinds yet so not much in the way of reads on his 3betting/flatting tendencies vs. a BTN open.

Is there enough value to be had that I should bet the river when he checks?

Poker Stars, $0.10/$0.25 No Limit Hold'em Cash, 6 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

Hero (BTN): $38.01 (152 bb)
SB: $66.43 (265.7 bb)
BB: $33.42 (133.7 bb)
UTG: $27.23 (108.9 bb)
MP: $5.48 (21.9 bb)
CO: $37.75 (151 bb)

Preflop: Hero is BTN with T:club: Q:spade:
3 folds, Hero raises to $0.50, SB folds, BB calls $0.25

Flop: ($1.10) T:spade: J:club: 6:heart: (2 players)
BB checks, Hero bets $0.50, BB raises to $1.50, Hero calls $1

Turn: ($4.10) 4:diamond: (2 players)
BB bets $2.20, Hero calls $2.20

River: ($8.50) Q:club: (2 players)
BB checks

Definitely bet river, probably around 7.50ish.

I'd prob check back flop though, feel like JTx are likely to get c/r and we aren't ever loving it.

Quite a large bet $7.50. I think I make it about $4 to get crying calls from his Jx. This river smashes our range, we have the nuts when we have AK.

As played I'm folding flop and turn, by the way. I doubt he's bluffing seeing as there's very little in our range that hates this flop.

I think 4 is too small imo. It's pretty clear he has showdown value here. I don't think he's necessarily folding KQ here for 6ish + bet, and def won't find a fold with JT for 6ish I don't think.

I'd fold KQ to a $7.50 bet here I think. Just looks too much like the nuts. I'd consider laying down JT to be honest. We rep AK when we are way behind the nuts. If we think this guy can hero fold a set then suddenly I like the idea of this size a lot more, but I think he's laying down AJ and KQ seeing as we can play AK exactly like this.

$4 might be too little. I'm gonna think about this some more. It's interesting trying to figure out optimal sizing here.