Thank you for the comments, everyone I have some news…
Now that I have had the chance to speak with the parish priest, I think my cousin misunderstood the deacon. As she was explaining the advise of the deacon, it sounded like he was inventing a stipulation that did not exist in the Code of Canon Law. And as much as I wish my cousin had a more lived faith, I was upset that the local church would unnecessarily delay the infant’s baptism over a made-up rule.
However, it became clear in my conversation with the priest that my cousin had misunderstood what was said. The deacon was merely explaining that since she had not been fully initiated as a Christian and could hardly call herself “Catholic,” there was no hope that the child be raised Catholic. The deacon encouraged her to first become fully initiated so that she could be a good Catholic parent to her son. And so, the deacon was not fabricating a rule, he was invoking an existing one.
My cousin, God bless her, could not understand all of this less, so I worry that she will react with hostility. That being said, the Church made the right call.