I also prefer betting that flop, for a number of reasons. You force him to define the hand, and if he doesn't check raise you, you get a free turn.

If you're worried about him having an A, but want to see a showdown, a flop and river bet are cheaper than the turn and river.

As it is, his turn and river bets offer you absolutely no indication of his hand-strength, as they are de rigueur because you showed weakness.