Can a husband be awarded joint custody despite severe mental illness?

There's no longer any such thing as "joint custody" in Florida. There is parental responsibility, which is either shared or sole, and timesharing. Generally, mental illness would not completely preclude someone from having parental responsibility or timesharing, but it is a question of fact and severity. Consult with an attorney to discuss the specifics of your situation and your options.

Surely, your matter requires much more than knowing that the husband has a mental illness which is severe. You can pay for a guardian ad litem to investigate and then make a recommendation to the court.