Does French administrative divorce count as "khulʿ" in Islamic law if it occurs at the wife's request due to her dislike of her husband, and the husband initially refuses "khulʿ" but then agrees to the French divorce, while refusing any mediation for annulment of the contract and considering the wife still his legal spouse?

Divorce pronounced by secular courts is not valid, nor is it valid merely by the husband's signature on its document. If divorce is agreed upon in exchange for compensation (kitchen equipment), this is a khul' (dissolution of marriage at the wife's instance), and its validity does not depend on its documentation by the court. Islamic centers act as Sharia (definition) courts in non-Islamic countries, so the sister can refer her case to one of them to consider whether khul' has occurred or not. If khul' has not occurred and the woman is still under her husband's marital authority, then efforts must be made to remove harm from her, either through reconciliation or divorce, and there is no harm in seeking assistance from secular courts for enforcement. It is not permissible for the husband to harm his wife and leave her in limbo, and if he does not mend his ways and repent, his consequences may be dire. He should be advised gently and kindly and reminded of God.