Is the money that the husband takes back from his wife, after he paid it to her in exchange for her leaving her job to raise their daughter, lawful for him, or is it impermissible for him to take it back Islamically?

Custody is a joint right between the parents as long as the marriage subsists. If they separate, custody belongs to the mother. The financial support of the children is obligatory upon the father by scholarly consensus, and it includes everything they need. It is permissible for the grandmother who has custody of her granddaughter to agree with the father on a fee for her custody if she left her job for that purpose. The grandmother’s undertaking of custody for her granddaughter in exchange for an agreed-upon fee is a binding rental contract for the father, and he must fulfill it, based on the words of Allah Almighty: "O you who have believed, fulfill [all] contracts." Returning the money to him is considered an act of kindness from the grandmother.