Dogs are kind of a weird case, from an animal population standpoint. On one hand, there are a ton of 'purebred' populations that undergo very specific selection

Dogs are kind of a weird case, from an animal population standpoint. On one hand, there are a ton of 'purebred' populations that undergo very specific selection and which are fairly genetically isolated from the rest of the species. These are your fighting dogs, your hunting dogs, your ratters, your herders, your companions, etc etc. On the other, there's this huge pool of castoffs and mutts, which aren't really undergoing any concerted deliberate selection at all. However, the castoffs from the specific breed dogs are constantly being shunted into the general population. So to answer your question in that context: Yes, it would be more than possible to take a population of dangerous fighting dogs and use them as the base stock for a pet breed, provided that the breeding program was well designed. However, the general mutt population will always have fighting dog traits mixed into it so long as there are fighting breeds around.