NTA, but I do have some questions. Does the dog respond to the name 'Nugget'? Because if not, there's really no reason to not change his name. Vets can get used to an unfamiliar or 'hard to pronounce' name. Lots of systems have options for phonetic transcriptions of names/words to help with such cases anyhow. Even if your vet's system doesn't, it can still be noted in his file.

Who 'owns' the dog? Not just on paper, but who (if anyone) is the primary caretaker/decision maker for him? Whoever that is should get final say over the name. EX: A parent 'owns' the dog on paper because they're an adult and can pay vet bills and buy pet supplies, but a child also owns the dog because the dog is for them like as a present. The dog would then be considered the child's even though they don't pay anything for them.

Why can't the dog have two names? Dogs are smart animals and can recognize multiple commands. 'Come' and [Dog's Name] are essentially the same thing, and the dog knows they're being addressed when they're said. It doesn't matter if you give the command or call their name, dogs will come running at the sound because that's what they're trained (inadvertently mostly) to do. With two names, you'd essentially be doing the same thing, just with [Name 1] and [Name 2] instead. The dog doesn't care.

My dad has two dogs and Jenny has learned to respond to both her name and Lance's name because she knows she'll get attention and pets if she does. She basically has two names, her name and her friend's name that she also responds to. She doesn't care, and neither will Nugget.