I really don't like the idea of playing a hand suboptimal just to disguise your hand. It's unfortunate that turns your hand faceup as JJ-KK, but the inherent value of your hand, relative to this villain's range, just doesn't justify a bet. He has a range consisting of worse hands that mostly fold to a bet, and better hands that never fold. Betting for deception / balance / so you can be a badass on the forum for c-betting 100% of your range, doesn't alter the facts of the situation at hand.

Now, many good regs will give you credit for c-betting a wide range of air on this flop and they will autocall the flop with 99+, and the flop is the most likely to street to get this value against them. That is a good reason to bet because it actually involves putting villain on a range and getting max value against that range. But I don't think this is the right opponent for a flop bet.

In general I think people are too afraid to play the check-call guessing game. It's not fun but sometimes every other way of playing the hand is really bad, so I guess we should try to get better at reading people's souls in these spots instead of taking clearly suboptimal lines to avoid the spots altogether (not necessarily referring to this specific hand because a flop bet does have some merit here).