A c/c on a non-Ace/King turn would be dependent on whether or not villain (1) has AK in his 3betting range to begin with, and (2) if he will 2-barrel with it with a high enough frequency to make the call +EV. If we use the range {KK+, AK} for villain, which btw I would weight more heavily towards being just KK+ [10/8 3betting UTG raise], and we can assume he (1) folds AK to a donk bet, and to a check/raise, and (2) that he cbets AK when we check, then this is really easily solvable with some EV calculations.

Which I will do after I take my dog out to go shit.

Edit: Also what you seem to be overlooking is that while the pot size will be the same whether we bet the flop, or villain bets the flop, the action we decide (checking or betting) will affect his range differently. If we bet, we assume he folds AK. However, if we check, we assume he cbets AK on the flop some % of the time.