I brought birth certificate with my first when in same situ (this was 10 years ago) but nobody questioned it.
I'd just bring birth certificate that shows the relation x
I had a picture of the birth certificate on my phone just in case when we traveled to the States in July, but no one questioned it! So I think you'll be fine but can bring it for peace of mind.
I’ve never had an issue with that with my daughter and tbh never even thought of needing anything with me. Cause she’s my daughter at the end of the day, is this something new? X