I don't know
Well, I'm a USC by birth and I don't have to show my birth certificate when I renew my passport, nor any other proof of citizenship than my old, expired passport. Why would anybody treat this girl differently?
I read a recent local case of a man who was illegally adopted as a small child. Apparently some 2nd cousin of his (apparent) mom had a baby (him) and gave it over to his (apparent) Mom. He has no birth certificate. There is no record of his birth. Even though he went to 12 years of public school, served in the army ect., he never had a birth certificate and somehow managed to live without it. Well, now, he, 57 years old, can't get a passport or prove his citizenship. He just wanted to go on a cruise.
But he just got tired of messing with it and took a chance. He recently left the country and reentered - carrying with him many pounds of paper detailing his long life in USA. They let him back in - after a couple of hours' delay at the port of entry.
But if your daughter can get this certificate of citenzenship without major hassles, it's probably worth it.
-Ocelot