You choose a working name just like breeders put on pedigrees but you don't need to have an affix, though you can should you wish to.
In the US, there are several agility venues that don't have purebred requirements, and my favorites are CPE and TDAA:Is a dog that isn't AKC or CKC registed allowed compete in agilty? My pup is purebreed but didn't come with any papers.