I'd be folding to the flop check raise, bunch of turns we hate, can't really call a 2nd barrel and out equity is not that good. I feel it's going to be too hard to win this hand and I don't think AQ is strong enough. AK I'd be calling the flop to take a turn.

While this check raise is really strange I don't think you can continue with AQ on this texture. Your bet is obviously good, but I'd just be giving up here.

3bet pot + checkraise is often sooooo strong. Looking forward, if I called the flop the only cards I would be happy calling again on are non-spade kings. That's just too little of the deck to want to continue so yeah I'd just fold flop. Luckily for you you should have plenty of better hands to continue with like a bunch of 2 pair and the nut straight.

Expecting villain to c bet his draws I don't mind checking the flop back.

As played it's the usual spot of beating none of villain's value range so folding looks best even when villain looks like he's not shy about check raising flops. There's going to be tons of hands you show up here with that can continue so AQ is really easy to forget about.