3b pot AQ on AKK vs fish-with-a-clue 50nl

pfr is 17-14 with 75% fold to 3b, caller from the blinds is 22-12 fish who wins at showdown with weird hands a bunch. I'm not sure he's calling a whole lot if i c-bet here, but i dunno, i cbet all my air in this spot obviously = ?? bet $4.50 as normal, or check it back?

PokerStars No-Limit Hold'em, $0.50 BB (9 handed) - PokerStars Converter Tool from FlopTurnRiver.com

UTG+1 ($56.15)
MP1 ($50)
MP2 ($51.95)
MP3 ($50)
Hero (CO) ($50)
Button ($84.15)
SB ($25.50)
BB ($49.55)
UTG ($79.85)

Preflop: Hero is CO with Qhttp://www.flopturnriver.com/pokerfo...lies/heart.gif, Ahttp://www.flopturnriver.com/pokerfo...es/diamond.gif
1 fold, UTG+1 bets $1.50, 3 folds, Hero raises to $5, 2 folds, BB calls $4.50, 1 fold

Flop: ($11.75) Khttp://www.flopturnriver.com/pokerfo...lies/spade.gif, Ahttp://www.flopturnriver.com/pokerfo...lies/heart.gif, Khttp://www.flopturnriver.com/pokerfo...es/diamond.gif (2 players)
BB checks,

You're not going to be getting much value from anything that isn't Ax here even if you check the flop.

I'd rather keep my range a little more balanced (though against this guy it probably doesn't matter but really if you don't cb AQ here a c/r bluff on this flop is a very profitable play for villain) by betting and probably getting 2 streets of value from Ax rather than check and maybe get 1 street from 99.

Those are my thoughts, where am I going wrong?

Isn't this one of those spots where the difference between betting or checking is negligible?

I just check. I don't wanna build a big pot, we're obv not gettin better to fold and a lot of worse hands that might put more money in on later streets will fold to a bet.

Betting this flop sucks after 3betting pre. You'd be basically telling him you didn't 3bet pre with AA/KK/AK, which leave hands that you fold to a c/r. And if villain expects you to bet flopped boats, he folds out anything you can expect value from.

I'd check flop, b/f if he checks over on turn, or call if he leads turn with every intention of flatting any river, unless he overbets, in which case I'd consider folding unless I have reason to think he overbets as a bluff often enough for AQ to be good here.

I don't think checking is terrible, but I would usually bet AA/KK/AK on this board, and always bet Kx and a lot of air so I don't see a problem with cbetting here.

If we check do we really get 1 street from 99? I sort of doubt it; delayed c-bet on this board is probably scarier than a flop bet. Does he turn 99 into a bluff if we check? Again, I doubt it.

If you're betting AA/KK/AK, then betting this flop is fine. I think I check this flop with AQ after 3betting because I check flopped nuts here against most villains.