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Playing weak FH on river IP vs loose opponent

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Playing weak FH on river IP vs loose opponent

Blinds: $0.50/$1.00 (5 Players) BB: $78.72
UTG: $95.40
CO: $101.07
BN: $243.33 (Hero)
SB: $61.46
Anonymous tables. BB's VPIP was over 90 with a much lower PFR, and I believe their AGG% was ~40 at this point. They may not have had any 3bets by this point, but I have their 3bet% as 6.3 in my DB now.
Preflop ($1.50) Hero is BN with 6 6 A 4
2 folds, Hero raises to $3.50, SB folds, BB calls $2.50
Button open on the looser side. Nbd.
Flop ($7.50) 6 A 8
BB bets $3.48, Hero raises to $13.94, BB calls $10.46
I don't think calling is an option here. Villain can have AA here, but more likely: pair + OESD, top two, or 88 given their flat preflop (not that AA is never just called preflop).

I raised to protect against straight draws for the most part; I don't think I'm earning many villain folds here, but if they do have 88 my range remains uncapped and there may be a non-zero chance of a fold from 88 if they put me on AA and only AA. I also didn't want to call and have BB take initiative to be able to barrel again on a straightening card that may not have actually given them a straight.

Also, I'm buying myself a river card a lot of the time.
Turn ($35.38) 6 A 8 4
BB checks, Hero checks
Really not a great card, and I'm liable to be check-raised by a significant portion of BB's range once they just call flop. I check to see if I can fill up.
River ($35.38) 6 A 8 4 8
BB
Full.

Trouble is, I don't actually beat much at this point. 88 has quads, A8 has a better FH... 57 may not call much of a river bet.

What's your plan for this river? How much money do you call to a BB bet (if anything)? (Or do you raise?) How much do you plan to bet if checked to (if anything)?

Comments on other streets also welcome.

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