Hand review request: Fold nut straight on river?

Did I play this fine and i'm just going to get stacked every time in this scenario? Or should I have folded the river over bet? Villain’s AF was 1.7
 
Yatahay Network - $0.02 NL - Holdem - 8 players
Hand converted by PokerTracker 4
UTG+1: 57 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 4)
MP: 93 BB (VPIP: 25.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 4)
MP+1: 443 BB (VPIP: 33.33, PFR: 26.67, 3Bet Preflop: 6.25, Hands: 181)
Hero (CO): 100 BB
BTN: 100 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 3)
SB: 117.5 BB (VPIP: 42.45, PFR: 1.89, 3Bet Preflop: 0.00, Hands: 107)
BB: 260.5 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: -, Hands: 5)
UTG: 100.5 BB (VPIP: 25.00, PFR: 25.00, 3Bet Preflop: -, Hands: 4)
SB posts SB 0.5 BB, BB posts BB 1 BB
Pre Flop: (pot: 1.5 BB) Hero has K♦ T♠
fold, fold, fold, fold, Hero raises to 4 BB, fold, SB calls 3.5 BB, fold
Flop: (9 BB, 2 players) J♠ Q♥ 7♦
SB checks, Hero bets 5 BB, SB calls 5 BB
Turn: (19 BB, 2 players) J♣
SB checks, Hero checks
River: (19 BB, 2 players) A♦
SB bets 108.5 BB and is all-in, Hero calls 91 BB and is all-in
Hero shows K♦ T♠ (Straight, Ace High)
(Pre 38%, Flop 38%, Turn 9%)
SB shows A♥ J♦ (Full House, Jacks full of Aces)
(Pre 62%, Flop 62%, Turn 91%)
SB wins 191 BB