She wouldn’t need to go through that. The child would be a US citizen because the mother was, though not natural born.
The laws on the books at the time of his birth hold that U.S. citizenship may only pass to a child born overseas to a U.S. citizen parent and non-citizen parent if the former was at least 19 years of age. Sen. Obama’s mother was only 18 at the time. Therefore, because U.S. citizenship could not legally be passed on to him, Obama could not be registered as a “natural born” citizen
I've read different accounts though on the laws regarding conferring citizenship to an underage mother's child, whether the birth occurred in or out of the country.
It is a well established fact the father was not a citizen, therefore zer0 is not eligible. But that would only count in a country where race doesn't trump the law.