Is it permissible to give an impoverished brother zakat from the money he is entitled to from his mother's inheritance, which has been relinquished to him for the future of his children?

You have done well by relinquishing your mother's inheritance to your brother. If zakat was obligatory before the gift, it is permissible to give it to your brother. However, if it became due after the money entered his possession, then zakat is obligatory upon him, and it is not permissible for him to spend it on himself or his father.