How would you handle this situation in proving a mother is unfit for custody?

I understand your issue but here is a thought. Why not try to get her some help for her anger issues? Do you know how damaging it will be for your child to hav no mom? If you are thinking of the best thing for your child, maybe look in another direction so your child can still see her maybe supervised. SHe may be a mess but maybe she can be helped.