Is it permissible for a father to divorce the mother or for them to officially separate so that he can pay off his debt through the government?

The man must offer sincere repentance for usurious transactions. If he repents and regrets, then it is permissible for him to request administrative separation to waive the usurious increase, while being obligated to return the principal amount to its owners, as Allah Almighty says: (And if you repent, you may have your principal – [thus] you do no wrong, nor are you wronged.)