All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, you should know that it is not permissible for a wife to ask her husband for divorce except in certain cases that we clarified in Fatwa 131953. Even if we assume that a wife has a sound reason to ask for divorce from her husband, she should not hasten to it, as the benefit might not always be in divorce; rather, she should be patient as much as she can, especially if they have children.
If divorce takes place, the mother has the right to custody of her children as long as she does not remarry as clarified in Fatwa 82128. If she remarries, custody moves to the female who is more deserving of the custody of the child among the female relatives according to the order mentioned by the jurists, which we clarified in Fatwa 84618.
On the other hand, the majority of the scholars are of the view that if one of the parents travels to another location to live and settle there, the custody is for the father. The father is even more entitled to custody if he is moving to a Muslim country as it is dangerous to the religion of the children to reside in a non-Muslim country where there are many causes for temptation, in addition to the freedom that has no limits. For more benefit on the danger of residing in a non-Muslim country to the upbringing of Muslim children, please refer to Fatwa 179175.
However, some scholars differentiated in this regard between an infant who is still breastfeeding and other children. They are of the view that the infant should not be taken away from his mother since he needs her care; this is due to the harm that is inflicted on the infant and his mother by taking him away from her, and because Islam has taken into consideration the interests of the child. After mentioning the statements of the scholars in this regard, Ibn Al-Qayyim said in Zaad Al-Ma’aad: “All these statements are not based upon evidence that comforts the heart as you see; the correct view is to look into the matter and consider the interests of the child and what is best for him, either keeping him where he already resides or moving him to another location. It is what is more beneficial for him and what better protects him that should be taken into account, and residing in the same location or moving to another one has no effect.”
As regards your question whether or not your husband has the right to refuse custody, then the answer to this question is based on the following issue over which the scholars differed: Is custody the right of the custodian or the right of the child? Ibn Al-Qayyim said: “The jurists differed in opinion over custody; is it a right for the custodian or a duty upon him? There are two opinions in this regard according to the Maaliki and Hanbali Schools of Jurisprudence; accordingly, can the one who has the right of custody renounce it? There are two opinions in this regard.”
In any case, the father is obligated to spend on the child whether the child is in his custody or in its mother's custody.
Finally, we advise you to try to reach a mutual understanding with your husband about what is better for your daughter and opting for what is best for her and renouncing your right for the benefit of your daughter. But if there is a dispute, then it is better to take the matter to an authority that is specialized in studying the matters of the Muslims like the Islamic Centers. This is of course if we assume that divorce has taken place, otherwise if it is possible to reconcile and continue the marital relationship, then this is more appropriate as we have already mentioned.
Allaah Knows best.