I actually like shoving the flop as a bluff. It would take a very strange range range to be ahead of you more than half of the time here (and you need him to fold 50% of the time to break even on the shove).
I don't think we can play QQ for value against a player like this, I don't mind it against a reg who you can make some sense of (even if that just means he has a well balanced range) but you'd need a very solid read to play this for value and I don't think that's likely here.
I mean, if you bet does he call with worse or give you credit for the A? In my opinion A's are the one card in a deck that fish will fold a mid pocket pair to. I'm just not comfortable betting this for value. We could of course check/call a shove, but for that you need to be confident that he bets/shoves with less than the A and I think we can very rarely be confident that he's going to put chips in often enough with weak enough hands to justify a call. If we check we only need to be ahead 1 time in 3 which against a someone with a balanced range is a super easy call, against a nit it's a fold, against a fish... I just can't see him thinking this hand through well enough to shove with less often enough, we're relying entirely on him spazzing out which I don't think even fish do >33% of the time. I think you're winning a small pot when you're ahead and losing a big one when you're behind, if you bet.
But I think a bluff shove is +EV on fold equity alone, plus we can still rely on him to make occasional calls with stuff we actually beat.