Housemate Is Deciding To Move Out And Wants Us To Reimburse This Month's Rent

In July, we filled two rooms in a house we're renting with two strangers. The rooms are directly across from each other and share a bathroom. We are all working professionals and thought the two occupants would be able to get along.

Very quickly, one of them became aloof and ignored the other. Good roommate's requests to clean actual bathroom messes made by the bad roommate were met with annoyance and completely ignored. Bad roommate then blocked good roommate and refused contact. Tensions grew and police reports were filed by both parties. Bad roommate didn't listen to our requests to be mature and civil, and has created a hostile environment consistently.

Recently, BR threw a towel at GR's face. A report was filed. Then, BR was caught stealing from GR with a discrete bathroom drawer camera (which is GR's space, not shared, and always closed). BR called the police and played victim, but they deemed laws weren't broken.

Two days ago we had a house meeting to tell them this isn't working. We don't see a resolution happening since we've asked for one for months. We verbally asked BR if it's possible for him to move out. He said yes and he already has a place lined up. He started moving his things out yesterday and wants us to sign a document saying in his words he's "deciding to leave" and would no longer be responsible for rent and utilities after 12/14 when he's fully moved out. I get that part, however, we never asked him to leave by any amount of time- just by the end of the month he's already paid for if possible. The document also says he wants us to reimburse him rent for the 17 days of December he won't be here. We didn't ask him to leave that early- are we still responsible to pay him his rent back out of our own pockets?

If we say no, can he reasonably stay until the lease is done like he's threatening?