The people are irrelevant.
But if they take the moon?
Have you played Future Risk?
There's a separate board for the Moon.
You take rockets up there and then rain nukes on the stupid terrans.
If you let these hypothetical malcontents declare independence only to find that 6 weeks later that they're trying to decide if they should join the Dominion or the Romulan Star Empire, it all just sounds too much like the Cuban Missile Crisis in Space to me.
Everyone lives where they're told, or petitions to live where they want and live where they're told until the local government confirms or denies their "request" to live where they want. I suppose if some turk vintner could make a good enough argument, like that all the Picards on earth are dead (did the wife survive?) that they should occupy Chateu Picard and rebegin production. The only thing stopping that is that Picard saved the Federation a couple times.
If the federation felt uncomfortable with the Lunal population they could disperase all 3 million to a hundred outter colonies, and restaff the moon with agreeable loyalist.