You cannot unlock Nintendogs in Super Smash Bros. Brawl. They are not playable characters, Assist Trophies, Poké Ball Pokémon, or Stage Elements.
If you want a dog fighter: Isabelle (Animal Crossing) in Ultimate uses a fishing rod/slingshot. Duck Hunt Duo (NES Zapper dog) in Smash 4/Ultimate. No Nintendog fighter exists.