AKs deep HU

Hey all, haven't posted in a while. This hand has been bugging me though so I'd like some outside opinions on it

Poker Stars, $0.02/$0.05 No Limit Hold'em Cash, 2 Players

Hero (BB): $12.50 (250 bb)
SB: $12.73 (254.6 bb)

Preflop: Hero is BB with K A
SB raises to $0.20, Hero raises to $0.55, SB calls $0.35

Flop: ($1.10) 3 A 7 (2 players)
Hero bets $0.85, SB calls $0.85

Turn: ($2.80) 2 (2 players)
Hero bets $2.25, SB raises to $11.33 and is all-in, Hero ??

Villain is 55/38, 22% 3bet, 3/6 fold to 3bet, 0/4 fold to CB, 3.0 aggro factor (these stats are like 95% from full table 6max) over 100ish hands

So basically my question here is does he ever ship any Ax I beat here or is it always 2pr, sets, and straights that I see when I call here? (I have a feeling I'm crushed most of the time but like I said, it's bugging me)