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10/20 deep - turn range play and river line?

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10/20 deep - turn range play and river line?

BN: $3393.46
SB: $6089.89
BB: $5217.20
HJ: $5599.11 (Hero)
CO: $2243.52
Villian is a laggy reg.
Preflop ($30.00) (5 Players)
Hero was dealt K A
Hero raises to $60, CO calls $60, BN folds, SB calls $50, BB folds
Flop ($200.00) 8 2 K (3 Players)
SB checks, Hero bets $162.75, CO calls $162.75, SB calls $162.75
Turn ($688.25) 8 2 K 8 (3 Players)
SB checks, Hero bets $253.89, CO folds, SB calls $253.89
I misread the hand and thought I was OOP to both.
How do we play our range on the 8? It's definitely not favourable to us, yet we want to find a way to realise our FD equity. Since this card doesn't completely flip the strength of our ranges, I think a 1/3 pot bet is good.

Should I just bet bigger with a more polarized range?

When BB calls, I don't expect him to have many 8x. Most fold the flop and some c/r turn.
River ($1196.03) 8 2 K 8 4 (2 Players)
SB bets $940, Hero folds
I'm not sure he's turning Kx with a club into a bluff here, but he may have read into my turn sizing.

If I raise, I need him to fold a flush. So I need to raise huge.

His flop c/r stat is somewhat high, so I expect him to c/r 22 100%.
It also takes a lot of K8 combos out of his range.

Folding seems fine too, as well as calling (if I think he can turn something into a bluff).
Final Pot
SB wins $1213.03

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