I don't think betting accomplishes much in this spot.
If you bet big, you isolate yourself against the strongest part of villain's range. Best case scenario, you get called by JJ or QQ, or a big draw, but what is your plan on the turn? Do you think villain will call another big bet with JJ on the turn, if it's a blank? If not, then you gained nothing by betting on the flop that a later bet on the turn or the river wouldn't also have accomplished. Conversely, what will you do if the turn is a 6, T, J, or a Q? Bet big again? And then what on the river?
And if you bet small on this flop, you are just begging to get check-raised. With AA you would have to call a raise (villain has a ton of draws in their range), and now you are suddenly playing for 250 BBs with just one pair. Not ideal.
If you check, villain now gets to the turn with all the same hands as in the scenario, where you bet big, and you will face the same decision if a scary turn card drop. But the pot will be smaller, which means that your mistakes will be less costly. In addition, villain will improve some amount of their unpaired hands which would have folded to a flop bet, which you can now get value from. And finally, you now have a relatively strong hand with which to fight back, should they decide to bluff.