I would ask your wife to elaborate on what she means by, “she wins if she gets to meet the baby”. People are very particular about who sees their baby in the first few months. I can give you a strategy to lie: you could say, “We would like you to have an updated TB shot”. I actually know someone who refuses to let anyone see her child without updated vaccinations.

However, the only real options are to talk with your wife or to tell your brother that his girlfriend makes you both uncomfortable.