Why can’t my 13-year-old brother-in-law be a godfather if he’s been confirmed?

Confirmation is not the only requirement for a person to be a godparent. Canon law also requires that the person be at least 16, unless a different age is stipulated by the bishop or unless the minister of baptism believes that there is just reason to make an exception (Code of Canon Law 874). Apparently your priest does not consider the reasons you’ve given so far to be just reason to make an exception for your brother-in-law.

One of the reasons that there is an age requirement for godparents is that the godparents are expected to assist the parents in forming the child to “live a Christian life befitting the baptized and faithfully to fulfill the duties inherent in baptism” (CIC 872). A 13-year-old is himself still being formed in the faith and may not be ready for the responsibility of participating in the spiritual formation of another.