What is the legitimate Islamic solution to the problem of my mother refusing to move in with me and my wife, despite their financial need, because of her hatred for my wife, and must I continue to incur debt to cover their increasing expenses?

We have previously explained that among the conditions for the obligation of kinship maintenance is that there must be an surplus beyond the maintainer's own expenses and those of his dependents. If you do not find a surplus from your expenses and those of your family to spend on them, then you are not obligated to incur debt to provide for them.

Strive to convince your mother to live with your wife in one house, such that each of them has an independent section if necessary. supplicate frequently for the matter to be eased, and try to appeal to your mother's compassion by explaining your financial hardship, and seek help from those close to her.