Praise be to Allah.
In the case of khula‘, it is not stipulated that the word of divorce be uttered; it is an annulment and not a divorce (talaaq) according to the correct opinion. When khula‘ is done, the wife is irrevocably divorced, and the husband does not have the right to take her back except with a new marriage contract.
Khula‘ may be given in return for financial payment, such as if the wife returns the mahr to the husband or gives up some of the delayed portion of the mahr. See the answer to questions no. 126444 and 133859
The qaadi may oblige the husband to divorce his wife by talaaq or by khula‘, if he finds cause to do so, such as if the wife is being harmed. If the khula‘ or divorce has been done through the court, the husband has no choice but to refer back to the court and try to convince the court to overturn its decision or to get an official fatwa from the relevant department in his country stating that the khula‘ is invalid, and submit this fatwa to the court.
And Allah knows best.