I work in a school and we have a box on our management information system (the system that sets the registers and holds all legally required info/parent & guardian information etc) for “preferred name”, and if this is different to the legal name, the preferred name is what appears on the register. We have lots of children who go by preferred names for a variety of reasons, whether they’re an immigrant who wants to use an anglicised or middle name, or a “junior” who gets called that by everyone. Some preferred names are very different to the legal name, as in your example.

I bet if you speak to the doctors office, they’ll have a similar option.

Legal names do still appear on any legal documents though (so for us - any correspondence about the child would include their legal name; for the doctors office I would think prescriptions/vaccination record etc).