Germany may offer the child citizenship, but as a child born overseas to a family on official duty, she is a native born natural US citizen.
The US does not recognize duality. Either you are a US citizen or you are not.
It also depends on where the child was born- if in a US military hospital on US installation, then Germany has no record of her birth, if in a German facility attended by a German physician, then she indeed may have a German BC, but that BC will specify the citizenship of her parents. If the latter, the parents needed to file ( the hospital probably did I while they were there) a form to the US Embassy reporting the “birth of a US Child abroad”, beginning the US State issuance of a BC.
My son was born in a similar situation, but in a US facility, he searched the German records for any reference to his birth in West Germany, there is none. He is a US citizen naturally, by virtue of our official duty there. He has a US State Department BC.
I wonder whether the “dual citizenship” issue might be the ticket on which this turns. We say we don’t recognize dual citizenship, but how is that really enforced? If someone is a U.S. citizen by birth, and another country allows them to obtain citizenship without renouncing their U.S. citizenship, as of now, what’s to stop them from doing so? But if the rule is (exclusively) subject to the jurisdiction of, then if someone is born here a citizen of another country, then that means they are not citizens here. Maybe.
The US is neutral on dual citizenship. There is no law opposing it.
As just a point of my own curiosity, when asked what state he was born in, what does he say?