Apologies! It looks like this is being dealt with on our Facebook Forum – https://www.facebook.com/groups/BritishinAustria/posts/1680146649129915/
As advised to your wife. You should go back to them and respectfully ask them to check the rules with their superiors at the BMI. Here is the Austrian Federal Government site –
“Wann und wo müssen meine Familienangehörigen den Antrag stellen?
Die Familienangehörigen – bei Kindern unter 14 Jahre die gesetzliche Vertretung – müssen den Antrag persönlich spätestens innerhalb von 3 Monaten ab Einreise in Österreich stellen. Bei Geburt in Österreich zählt diese als Einreise.”
https://www.bundeskanzleramt.gv.at/themen/brexit/aufenthalt-zugang-zum-arbeitsmarkt.html
and in English
“When and where do my family members need to apply?
Each family member – in the case of children under 14 years of age the legal representative – must submit the application in person within 3 months of entry into Austria at the latest. In case of birth in Austria, this counts as entry.”
https://www.bundeskanzleramt.gv.at/en/topics/brexit/residency-and-access-to-the-labour-market.html
Basically if you do it within 3 months of the birth it should be fine.
I believe your wife is going to drop Mike an email as we have issues with Salzburg Stadt a few times.