Just my opinion, not an attack of any sort as I am a live-and-let-live kinda guy, but I vote no on animal’s having souls.
My reasoning is that if animals had souls, God would not have told Adam that they were all placed on this earth for his use. I know that this is a very broad and generalized statement, but you get the gist, Adam was allowed to use the animals (responsibly) as food and as beasts of burden.
I am of the opinion that God would not subject a being with a celestial soul to being destined to be dinner or slave labor/indentured servitude.
Again, just my opinion, which is free, and worth every penny.
The animals are incapable of some things, such as moral choice as we know it. A bad dog is a factor of its breeding and environment.
The term, nephesh, generally meaning soul, is used in the creation story with respect to a large segment of the animal kingdom. Tying it to breathing doesn’t quite work, because this term is also used of fish.
A slavery or servitude would be wrong if moral choice were possible of the animals. As it were, we are still bidden to regard our beasts’ lives (or as commonly rendered in bibles, be kind to them). But an eternally remaining part would be free of whatever that slavery or servitude was.