What is more befitting for a woman who despises her husband and falls short in fulfilling his rights: should she remain with him for the sake of the children, despite his lack of interest in raising them Islamically in Western countries, or should she seek a Khul' (divorce initiated by the wife) and emigrate to a Muslim country to raise her children herself?

It is preferable for a woman to remain with her husband, even if she dislikes him due to his disregard for Allah's law. In such a case, she should invite him [to guidance] and direct him with wisdom and good counsel, and cooperate with him for the continuation of their marriage and the upbringing of their children. If her dislike for him stems only from emotional aversion, then she should fear Allah and be patient, for (perhaps you dislike something while it is good for you). However, if all avenues of reform are exhausted and she is unable to fulfill the limits of Allah, then there is no blame upon her in seeking khul' (divorce at the wife's instance by returning the dower), and this is the last resort.