$2 NLHE 6-max: Double barreling vs fish

Villain is 83/12/6. Can we bet here and X/F river if the draw misses?

poker stars $0.01/$0.02 No Limit Hold'em - 6 players - View hand 2219993
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Hero (BB): $2.97
UTG: $2.28
MP: $3.74
CO: $2.48
BTN: $3.20
SB: $1.00

Pre Flop: ($0.03) Hero is BB with K K
2 folds, CO calls $0.02, 2 folds, Hero raises to $0.10, CO calls $0.08

Flop: ($0.21) 7 7 J (2 players)
Hero bets $0.14, CO calls $0.14

Turn: ($0.49) A (2 players)
Hero bets $0.28, CO calls $0.28

River: ($1.05) 4 (2 players)
Hero checks, CO checks

you could go for a blocking bet/ thin value on the river, something like $0.50

First, I'm going near pot on flop and turn. We want to get max value out of this super station.

Second, I'm kinda torn on river tbh. I like your check, as I'm assuming you were thinking he had a draw which is obv not going to call a river bet, but might try to fire w/air.

I can see the argument for value betting v Jx, but I just think villain will show up w/hands like Ax spades too often or a very oddly played 7x for us to VB and has too much SDV w/those hands to ever fold to a river bet. By betting river, I think we allow villain to fold out all his air that missed + call w/hands that beat us.

Ya, now that I think about it more, I really like check/calling river if you think he's capable of bluffing a missed draw, which since he's 83/16 i'm assuming he was.