NL10 Butchered flopped straight

[flameboy]

Full Tilt Poker - No Limit Hold'em Cash Game - $0.05/$0.10 Blinds - 5 Players - (LegoPoker HH Converter)

Hero (SB): $14.05
BB: $13.60
UTG: $9.80
CO: $19.15
BTN: $10.30

Preflop: Hero is dealt T[img]/images/graemlins/heart.gif[/img] Q[img]/images/graemlins/spade.gif[/img] (5 Players)
2 folds, <font color="red">BTN raises to $0.35</font>, Hero calls $0.30, BB calls $0.25

Flop: ($1.05) 8[img]/images/graemlins/diamond.gif[/img] 9[img]/images/graemlins/club.gif[/img] J[img]/images/graemlins/club.gif[/img] (3 Players)
Hero checks, BB checks, BTN checks

Turn: ($1.05) 3[img]/images/graemlins/club.gif[/img] (3 Players)
Hero checks, <font color="red">BB bets $0.50</font>, BTN folds, Hero calls $0.50

River: ($2.05) 9[img]/images/graemlins/diamond.gif[/img] (2 Players)
Hero checks, <font color="red">BB bets $2.00</font>, Hero ???

Button is 40/35 and has raised my blind every time so far - and then shown a tendency to stack off 3-barreling my top-pair check/calls with nothing to the river.

BB is 15/2 [0.03/0.67/2.0/3.0] but plays solid post flop. I'm pretty sure he wouldn't play TPTK like that.

Preflop and Flop is standard against this button raiser, but I'm not sure about the turn and the river against the BB.