You showed strenght PF, you need at least a C-bet.
I have range advantage but lack nut advantage (cutoff could have sets).
You're expected to bet, and a check will give him a free card, that's not good. Every bet is a question you ask your opponent: "are you strong enough to call?". Here, an above average bet will give you important information. Make it expensive enough to make him think about the value (and possibly scare him), not enough to be pot commited later (which you shouldn't be right there with 250BB).
After you 3-bet (or a little more, like 3.5) him PF, a strong c-bet will show strenght and probably polarize your hand (Q♦Q♠+). If he check-raises you, now will be time to make a decision. He can have suited connectors (8♥9♠) and hit 2 pairs. Or a middle pair 8♥8♦and hit a set.
Or your bet will simply make his hand unworthy of a call: with air or K♣Q♥ or K♣J♦ for instance, he's behind any pair (that hit the flop or not) or any strong A♣
You can keep the pot small with a check or small bet. But then? Another free card on the turn? What if another scary card shows up? Your PF 3-bet and C-bet can also make him think you hold T♣T♠ or J♣J♠ and fear switches sides.
He's the CO, he may play a wide range of hands. He may think that you're trying and steal the pot and call your 3-bet PF. But then? He needs to hit the flop as badly as you need to polarize his hand. You were the aggressor PF, stand your ground and fire a second barrel.
In your case, I want an answer, and I don't give any free card.
If he folds, I'd rather win a small pot than lose a big one.