I would register. Every breeder I've used required the puppy to be registered, and their name had to began with the kennel name. My poodle's name had to have kennel name, then a name that began with a "Y" because it was from the "Y" litter. So, her registered name is Tiara Your my Jasmine. Her call name is Jazzy.
Check with your breeder to see if they have any requirements.
Check with your breeder to see if they have any requirements.