Is it permissible for a woman to divorce her husband in order to restore the marital bond to three pronouncements of divorce after two have already occurred, and is this considered toying with the religion of Allah, like the muḥallil?

Khul' (divorce at the instance of the wife) does not annul a previous divorce. There is nothing in the statements of the scholars indicating that khul' invalidates prior divorces. Both spouses should preserve family stability through mutual respect, overlooking mistakes, and each fulfilling their duties towards the other.