If the interview letter asks for a birth certificate, then I would bring it. How far ahead is the interview?
Sometimes BC do not take that long, and it is a good document to have. What if she loses her passport and needs proof of citizenship asap? Google Vital records of the State she was born. Sometimes you're able to request it online.
For my own interview (I applied on the three year rule) I brought my husband, his birth certificate, his passport, my passports, a physical copy of all the documents I'd submitted with the application, plus new, updated evidence. Officer just asked for my DL and my green cards. But I feel it's better to have what they ask for in the letter and let them handle it however they want it.