You can 4bet/call pre.
Given the preflop line, you're towards the bottom of your perceived range. The bottom of your perceived range seems like AQ, AJ and QQ. You have all the sets and full houses in your perceived range, and even a straight sometimes.
When you don't bet the flop, you cap that range to the straight (feasible to check the nuts here imo), and the weaker stuff. Sets probably bet flop. Most of the time, anyways.
That means you have 4 combos of something that's happy, 6 combos of something that's not that happy, and 3 combos of something that's unhappy.
Now, if he has KK, he can bet two streets here because of combinatorics. If he has AA, he's a little more uncertain. Doesn't mean he won't bet, but it makes him a little more uncertain.
The way you played it, I think you played it well.
Oh, and move down fish.