A minor child cannot renounce his birthright U.S. citizenship. Nor can a parent renounce it for him.
At the same time, the minor can be considered a citizen or a national of another country (e.g., Indonesia). But he does not lose his claim on U.S. citizenship upon reaching his majority.
So, it is quite conceivable that State Department documents might identify him as an Indonesian national -- but it does not change the fact of his birthright claim upon reaching his majority.
Note: A person making such a birthright claim is NOT considered a "naturalized citizen". They are a U.S. citizen from birth.
You can repeat it but it doesn't make it true. which it ain't. There's a document from the State Department posted on this thread proving this.
the fact of his birthright claim
His claim is more statutory than "birthright," assuming his biological and legal father is the British subject he says it is.