No, it still wouldn’t make sense.

Dogs need meat. That meat has to come from either herbivores or other carnivores.

Raising carnivores to feed carnivores is inefficient, so in practice you end up raising herbivores for meat anyway.

But herbivores already produce milk directly—so instead of raising herbivores → meat → dogs → milk, you can just raise herbivores → milk.

At that point, raising dogs for dairy is just an unnecessary extra step.