I'd just go for a range bet there, limp-call range is hard to figure out but we should have enough of an advantage

Betsize on flop can't be right because when UTG shoves it's only 1.5 to win 8.5.
We don't need to go for a big sizing to get stacks in either if we had a stronger hand. 1/3rd pot would set up turn all in anyway.
And a smaller size might also make it more likely UTG calls with Tx. Although given it's a fish that might be the case anyway.

So why bet QQ instead of check? If UTG barely ever bluffs we don't want to check. Because we won't generate extra value from worse. And it's also unlikely that the unpaired hands are going to put in extra money of they catch a pair on turn or river. So checking to get value from a future worse made hand barely exists.
Denying the equity of pocket pairs, gutshots and maybe Kx seems worth it.
Plus range betting just makes your life easier.

Bet smaller

Betting range is all right but in that case it's pointless to bet this big. Think about what your goal is with your bet (or even your check if you choose that, I would btw).

What you don't want is for him to continue with only stronger hands and fold out all the weaker hands. Therefore any action we can take to shift this even a little bit is better. So you can like write a list or something with different kind of actions as an exercise and write down how that affects the range he's gonna stay in the hand with. For example if you'd bet pot, he will most probably fold everything but an Ax and a Tx. If you'd bet 1 bb he probably calls with a bunch of air hands that doesn't even have a draw. If you check, his whole flop range will stay in the hand.

This is just a useful thing to think trough I believe, in order to have this kind of thought process. Based on the above I would check but betting small can also be around the same EV (25-33% pot). I still prefer checking, he can make bigger mistakes with hands that would not even consider continuing.