There is no need for American law to "prevail" as those born subject to another country's nationality law are not birthright citizens per the 14th Amendment as written and intended.
The Supreme Court has loosened the 14th Amendment's requirements somewhat in ruling that children born in the U.S. to parents who were permanently domiciled in the U.S. at the time of the child's birth also confers birthright citizenship. - U.S. v. Wong Kim Ark
Those born in the U.S. to a temporary resident alien are not legally born U.S. citizens at all, let alone natural born citizens.
Yes, there is a reason for it to prevail. It is how dual citizenship works in the United States at least. This way the child is entitled to her rights as an American, and Americans give precedence to these rights over her rights and obligations as a British subject.