Is it permissible for a couple to agree that the wife waive a portion of her mahr in exchange for the perpetual custody of their infant, even if she remarries, and is it permissible for her to have his custody if she remarries?

It is permissible for the child to remain with the mother if the father waives custody, as the mother has the right to custody unless she remarries or has an impediment. If the mother remarries, custody transfers to someone else. Scholars have differed on who is most entitled to custody after the mother, and Ibn Taymiyyah and Ibn al-Qayyim preferred the father over the grandmother. If the parents agree that the father waives custody upon the mother's marriage, it is permissible for the child to remain with her, as Sheikh al-Islam Ibn Taymiyyah issued a fatwa permitting khulʿ (divorce at the wife's instance) in exchange for waiving rights and the mother taking custody of the child without claiming maintenance.