His 3bet calling range probably won't change with any amount <$2, so bigger is better.

On the flop I think he min-raises a wider range than you think (combo draws, air, maybe a hand like JJ to "see where he's at"), 1 king is out and we have AK so KK+ is discounted.

We can value shove the flop vs his calling range (59% equity).

66, AK, Ac9c, AcTc, AcJc, KT+, JcTc, Jc9c, Tc9c