Hi Lou
As far as I know, you wouldnt be able to register Alfie yourself, this is something only the breeder can do - within one year of the birth of the litter I think.
KC registration is mostly to ensure that the breeding adheres to the KC breed standard - for example, the dam can only be bred from a certain amount of times, after a certain age, up until a certain age, and after having had the appropriate health tests. All of which ensures the standard of the pups produced, thus maintaining the quality of the breed in general.
If you're content with Alfie as a family pet only, with no aspirations to show, then his not being registered (if you're sure of the pedigree) won't matter at all. However, I dont wish to worry you unnecessarily, but sometimes, a breeder may stall on the registration because the litter might not correspond to KC rules. So for this reason alone, I'd pursue the registration with your breeder.