If there is no separation agreement that involves custody then they still share equal rights to the child. He is within his rights to file for custody and to keep her here in NC. If the divorce is filed, she will have 30 days to respond and then a hearing will be scheduled if there is no response. Your brother would then go before the court, state that they have been separated since xxxx date and divorce will be granted. She does not have to sign the papers for the divorce to be granted.
If she does not want to sell the marital home, then she should be required to “buy him out” so to speak. They are both entitled to 1/2 the equity and if he wants out that may be the only way to get it. This way she can do whatever she wants with it afterwards. IMHO, having rental property in NC while you live in CA is just not good business unless you can afford to hire someone close to sort of watch out for the place and upkeep. Is anyone still living in the marital home?
He needs to consult an attorney first, since no agreement has been signed all marital assets are up for negotiations, and file for ED. He should also file for custody. He can always settle for joint custody afterwards, but in order to keep her in NC he will need to have it filed.