What is the ruling on practicing law in the absence of Sharia courts and governance by man-made laws, and is it permissible to accept a suitor who works as a lawyer under these circumstances?

It is not permissible for a Muslim to work as a lawyer in secular courts, because ruling by other than the Shari'ah of Allah Almighty is disbelief, injustice, and wickedness. It is not permissible to work in or with these courts unless it is for the purpose of achieving some benefits or reducing harms, such as defending the oppressed, or establishing proof against those who seek judgment from Taghut (tyranny/false deities), or filling a position that a tyrannical person might occupy. Otherwise, it would be cooperation in sin and transgression.

As for the suitor, if his religion and character are pleasing to you, he should not be rejected merely because he works as a lawyer in those courts, until the ruling on working in them is explained to him. If it becomes clear that his work is impermissible and he insists on it, then he should not be married, because his religion is not pleasing, and his earnings are illicit.