Praise be to Allah.
If a woman is divorced according to sharee‘ah, as mentioned, and she wants to document that in a non-Islamic court, there is nothing wrong with her doing that if there is no sharee‘ah court in her country, because documenting the divorce protects her rights, prevents the possibility of disputes in the future and enables her to marry again, as well as other benefits.
In a final statement issued by the second conference of the Shar‘i Council of Fuqaha’ in America, which was held in Copenhagen, Denmark, with the Muslim World League, 4-7 Jumaada al-Oola 1425 AH/22-25 January 2004, it says:
If a man divorces his wife according to sharee‘ah, there is nothing wrong with documenting it in the non-Islamic court, but if the spouses have a dispute concerning the divorce, the Islamic centres take the place of the sharee‘ah court if there is no sharee‘ah court, after completing the necessary legal procedures. Resorting to man-made laws to end the marriage from a legal point of view does not, by itself, mean that the marriage has ended from a shar‘i point of view. End quote.
For more information, please see the answer to question no. 127179
And Allah knows best.