Is it Permissble to Read Adhkar on Behalf of One's Child?

This is not permissible. This is going extreme in the religion, and the Messenger ﷺ used to say: "Doom is for those who go extreme." It is enough for a Muslim to do what Allāh has legislated for him or made obligatory upon him, like these Adhkār, for himself. As for doing them on behalf of others, then no, especially when that person is one who is excused, for he is a small child who is not mukallaf (held accountable). Therefore, it is not permissible for this mother to read these statements on behalf of her child, who is not mukallaf.