I do have a slightly serious question for OC, though. I went back and read the post carefully (something I hardly ever do, I know)... if you don't call non-working dogs "fake," what do you call them? You say working dogs are real... and non-working dogs are the opposite of working dogs... Dictionary.com defines the antonyms of "real" as:
"false, imaginary, imitation, unreal."
and the antonyms of "fake" as:
"authentic, bona fide, genuine, original, real."
As far as the words you're using go, I don't see that you have much choice of a definition for the non-working dog.