Is it permissible for a married woman to ask for independent housing separate from her husband's family, especially if she is suffering from problems with her mother-in-law who lives with them, knowing that her husband refuses this request and prefers divorce over it?

You have done well in your treatment of your husband’s mother, so continue to repel evil with good and avoid discord. Beware of being a cause for corrupting your husband’s relationship with his mother, for she has a great right over him, and obeying her is a cause for him entering Paradise. Inform your husband of his mother’s mistakes so that he may address them with wisdom.