Perhaps the OP is referring to the same child as in
this thread, though the earlier thread states that the child was born in Canada while above the OP states that the child was born in California. So unclear if this is the same child or not.
If it is the same child, presumably the child is over the age of 13 and it is expected of a child over the age of 13 to have lived at least two years in the UK before registering as a British citizen. So the child should enter the UK on an ILE visa (as the British citizen parent is claiming sole responsibility) and reside in the UK for two years, attending school here and then register as a British citizen after two years.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.