Struggling w BB defence against tight villain

NL Holdem 0.10(BB)
SB ($19.82) [VPIP: 25.1% | PFR: 18.6% | AGG: 35.8% | Hands: 222]
HERO ($10.38) [VPIP: 23.1% | PFR: 18% | AGG: 43.8% | Flop Agg: 47% | Turn Agg: 47.7% | 3Bet: 9.2% | 4Bet: 9.6% | Cold Call: 10.5% | Hands: 9436]
HJ ($10) [VPIP: 27.7% | PFR: 19.1% | AGG: 30.4% | Hands: 50]
CO ($9.28) [VPIP: 53.8% | PFR: 41.5% | AGG: 23.1% | Hands: 65]
BTN ($10) [VPIP: 19.1% | PFR: 12.8% | AGG: 46.2% | Flop Agg: 33.3% | Turn Agg: 40% | 3Bet: 8.3% | 4Bet: 0% | Hands: 48]

Dealt to Hero: K♥ J♣

HJ Folds, CO Folds, BTN Raises To $0.20, SB Folds, HERO Calls $0.10

Hero SPR on Flop: [21.78 effective]
Flop ($0.45): T♦ 9♥ K♣
HERO Checks, BTN Bets $0.34 (Rem. Stack: $9.46), HERO Calls $0.34 (Rem. Stack: $9.84)

Turn ($1.13): T♦ 9♥ K♣ A♦
HERO Checks, BTN Bets $1.35 (Rem. Stack: $8.11), HERO Folds

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• Pre - Against most viillains, I have a 3bet/fold strategy in ths spot. But against this viillain, I don't think I get a lot of folds and don't love the idea of playing a big pot OOP, so I call
  
• Flop - I check most of my range. Looking back, maybe a x/r and fold to 3bet maybe better than x/c? I don't think I'm ever leading on this board.
  
• Turn - In game I didn't think very far beyond "villain is nit, sizing is just thick value all day". Looking back, I don't think would change that line of thinking. Villain maybe has (KQd, 98d) as bluffs? And I'm not sure he's folding bottom of his value range to a bluff (AK, AQ, 99)
  
• Solver also has villain supposed to be betting small with most of his range on the turn. On a board this wet, why would that be?