Hit my hand, but river x/shove is usually the nuts right? His combo's that hit a FH on this board are so limited, 77/KJ, but both are certainly in his flatting range and I can't discount them entirely when the action is saying he got it?

Fold to river shove or sigh call?

PokerStars - $0.10 NL FAST (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

BTN: 197 BB (VPIP: 20.00, PFR: 20.00, 3Bet Preflop: 0.00, Hands: 10)
SB: 107.7 BB
BB: 89.8 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: -, Hands: 2)
UTG: 100 BB
Hero (MP): 100 BB
CO: 125.8 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 7)

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has 3 A

fold, Hero raises to 3 BB, fold, fold, SB calls 2.5 BB, fold

Flop: (7 BB, 2 players) 7 K K
SB checks, Hero bets 4 BB, SB calls 4 BB

Turn: (15 BB, 2 players) J
SB checks, Hero bets 8.5 BB, SB calls 8.5 BB

River: (32 BB, 2 players) 5
SB checks, Hero bets 18.3 BB, SB raises to 92.2 BB and is all-in, Hero ?

I don't see how this is not a full house.
Fold IMO.