NL10 Top pair 2nd kicker vs CS fish on busted draw runout

PokerStars Zoom, Hold'em No Limit - $0.05/$0.10 - 6 players

UTG: $9.28 (93 bb)
MP (Hero): $19.72 (197 bb)
CO: $4.38 (44 bb)
BU: $26.58 (266 bb)
SB: $22.82 (228 bb)
BB: $9.12 (91 bb)

Pre-Flop: ($0.15) Hero is MP with T♣ K♦
1 fold, Hero raises to $0.25, 3 players fold, BB calls $0.15

Flop: ($0.55) T♠ 3♥ 7♥ (2 players)
BB checks, Hero bets $0.41, BB calls $0.41

Turn: ($1.37) 3♦ (2 players)
BB checks, Hero bets $0.76, BB calls $0.76

River: ($2.89) 4♣ (2 players)
BB checks, Hero bets $1.93, BB calls $1.93

Total pot: $6.75 (Rake: $0.30)

Showdown:
MP (Hero) shows T♣ K♦ (two pair, Tens and Threes)
(Equity - Pre-Flop: 68%, Flop: 82%, Turn: 95%, River: 100%)

BB mucks 2♠ 7♦ (two pair, Sevens and Threes)
(Equity - Pre-Flop: 32%, Flop: 18%, Turn: 5%, River: 0%)

MP (Hero) wins $6.45

Villain was 35/17/8/2.3 AF 28 WTSD on 96 hands. Would call that half a fish. Was definitely taken away a bit when I saw his hand at showdown. Didn't think this was in the BB def range, but apparently it is. Sure vpip/pfr gap is big, but that seems a bit much.

Since I must assume that he is a bit fishy anyway and at least not overly aggressive with his 2.3 AF I thought I just bet 3 times here on the brick run-outs. Didn't expect his range and also not that he would be so stationy with 2nd pair no kicker.

What you think about my bet sizes? Can I adjust something here in this spot against this kind of villain type?