Take a look at your record and make certain that there is an e-ticket number associated with each passenger, including the infant. So long as each passenger has an e-ticket, especially the infant with name spelled correctly, this is a non-issue. If there is a reservation but no e-ticket for the non-existent infant, forget it. If there is a reservation and an e-ticket for the non-existent infant, just make certain that you were not charged more than you should have been and then forget it.

But, if your infant does not have an e-ticket and Baby Non-Existent does, that is potentially problematic depending on just how different the name is. Having nothing to do with any of the ordinary issues, there is heightened scrutiny for human trafficking purposes and there is a risk of Infant B traveling on Infant A's papers. So, rather than dealing with a lot of nastiness at departure, best to get the record cleaned up. BA is being obstinate, get AA & BA on a 3-way call and get it straightened out with you on the phone.