KK vs check/raise on dry flop

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KK vs check/raise on dry flop

Blinds: t150/t300 (5 Players) UTG: 25,674 (Hero)
CO: 21,011
BN: 14,959
SB: 47,982
BB: 17,893
SB: 159.94 BB (VPIP: 23.44, PFR: 17.82, 3Bet Preflop: 11.42, Hands: 678)
Preflop (450) Hero is UTG with K K
Hero raises to 750, 2 folds, SB calls 600, BB folds
Flop (2,000) 3 2 8
SB checks, Hero bets 676, SB raises to 2,014, Hero calls 1,338
Turn (6,028) 3 2 8 7
SB checks, Hero bets 3,760, SB calls 3,760
River (13,548) 3 2 8 7 J
SB checks, Hero bets 8,754, SB raises to 41,418 and is all in, Hero calls 10,356 and is all in

Villain is a reg, no reads. OTF, I would assume he is only raising sets, small and mid PP for protection, some of his FD and maybe 8x, so we have an easy call. I would also assume he keeps betting all his sets and 87s almost 100% on that turn, so I decided to bet again to get value from his FD, 99, TT and random 8x.

I think I completely butchered the river: against that hands that I am putting him on, I should probably bet more like 1/3 pot. As played, I could not think of a single hand that would shove river after checking turn, so I called.

Exploitatively, can we get away with folding this river? Although the way he played the hand makes no sense, I do think 90% of regs would not shove that river as a bluff with that SPR, plus it is still possible that some % of the time he still has sets in his range.

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