Considering that your kid has a valid US passport he is entitled to the Schengen entry as well. Since you are the parents you may need your marriage certificate and the baby’s birth certificate; only for them to verify that she is your kid (they want to be sure you are the legal parents/guardians).
As long as she is a US passport holder (passport valid for 6 months) she won’t be denied entry.
As a reference you can read this:
https://travel.state.gov/content/passports/en/go/schengen-fact-sheet.html
https://en.wikipedia.org/wiki/Visa_requirements_for_United_States_citizens
But if she has been in Spain for more than 90 days and doesn’t have any Spain resident documentation i.e. on the passport, a resident card etc then you should wait for it. The reason is she may asked to leave the Schengen area and enter it again which may pose a problem for you.
If she isn’t 90 days old yet then there would be no issue.
I was a in the Schengen area as a tourist and was offered employment in the Netherlands within my 6-month stay. So during my work stay I wanted to go for a weeks vacation (while my employer applied for a visa conversion). I was told at the Netherlands exit immigration that I wont be allowed to travel within the Schengen area after 1 week (because the 6 months were coming to an end) and I would need to exit unless I have a resident visa of any Schengen country.
Credit:stackoverflow.com‘
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