Going for a check raise on the flop wasn't terrible. So you didn't miss any value there.

I think you may have been able to get your opponent to call your bet on the river though if you'd have raised more on the turn.

I'm not certain, though, whether or not donking out on the flop has more value.

Sometimes your opponents AK misses, and he's just not going to put any more money into the pot. This is one of those cases IMO.