You're correct that renaming the adapter in Hyper-V networking will have no affect on the client. I don't believe it's a name that can be edited but I could be wrong.
Hopefully someone will have a better answer but I don't modify the registry for these types of things unless I know I've got all the items that need to be changed or, as might be in your case, it's purely a test machine.
It's worth noting that if this were a real adapter and you were supplying drivers you would edit the name before installing the driver and that name would be the name used by Windows. Since Hyper-V supplies the drivers via the integration tools you don't have that option.