enough combos to call to justify a bet vs x?

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enough combos to call to justify a bet vs x?

Blinds: $0.01/$0.02 (6 Players) BN: $2.10
SB: $4.81 (Hero)
BB: $2.10
UTG: $6.87
MP: $2.23
CO: $2.05
Preflop ($0.03) Hero is SB with A A
UTG folds, MP raises to $0.06, 2 folds, Hero raises to $0.21, BB folds, MP calls $0.15
Opponent is 15/12 w 2% 3bet. If the stats represent his true frequency, then his calling range is obviously very narrow.
Flop ($0.44) 9 A 4
Hero bets $0.17, MP calls $0.17
Nevertheless this flop I'm betting with all hands I 3bet vs him.
Turn ($0.78) 9 A 4 9
Hero bets $0.38

I see 32 combos left that we want to valuebet against KK-TT, AsKs, AsQs, AsKd, AsKh, AsKc, AsQd, AsQh, AsQc, maybe 99 making it 3 more combos.
I will never bet this turn with my QQ, JJ and vs him probably not with KK. Also not sure if I should with block bets.
Disregarding the range protection and balancedness (since 2NL), nevertheless it still seems to be correct to check in this spot. Am i right?
Are there any reasons not to go x-c, x/shove?

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