Is there a basis in the Shariah for a husband who pronounces Talaaq Baa-in to obtain the approval of a non-Muslim court declaring the Talaaq to be val

Answer

There is absolutely no basis for this. The Talaaq Baa-in comes into effect immediately the husband pronounces it. The Nikah is immediately and irrevocably terminated with the utterance of one Talaaq Baa-in. Neither is the approval of a non-Muslim Court or of a Muslim Court necessary for the validity of the Talaaq, nor can the disapproval of any court invalidate the Talaaq issued by the husband.