Bet-fold Opportunity? AK

Rush poker, no hands on villain. I think checking was a mistake: is this a good bet-fold opportunity?
When he bets I have him on a range like this: QQ, TT, AQ, ATs, KJs, QKs, QJs...I think JJ would check behind with its showdown value, and I don't see him stabbing with air.
So bet-fold? And as played?
Thanks

$0.05/$0.10 No Limit Holdem
6 Players
Hand Conversion Powered by weaktight.com
Stacks:
UTG ($18.38)
UTG+1 ($10.44)
CO ($11.26)
BTN ($29.96)
SB Hero ($18.52)
BB ($12.23)
Pre-Flop: ($0.15, 6 players) Hero is SB :ad: :kh:
1 fold, Utg+1 raises to $0.35, 2 folds, Hero raises to $1.20, 1 fold, UTG+1 calls $0.85
Flop: :qh: :tc: :ac: ($2.50, 2 players)
Hero checks, UTG+1 bets $2

Problem I see is... what hands does he think you can have here after you check? He's narrowed your range to just two hands... KK and AK, and both can be pushed off this flop under the circumstances. JJ is definitely in his range here, as is AK, and perhaps 99 and smaller pairs, since your pf 3bet prices in pairs looking to hit set. I think I prefer a b/f line, just because we can't stand much heat if we allow him to control the pot; I imagine you folded to this bet. If he has AQ TT or QQ, he might just call because AA (and QQ if he doesn't have it) is in our range, and so we might see another card under our terms with a nut draw. I don't think it's QT KJ KQ very often, these would be moronic pre flop calls. Betting might have AK folding, and certainly any small pairs he decided to continue with pre flop that he might find a bet with after we check. I think we're probably beat, but by checking we're accepting this without a fight. B/f for me.