I was wondering the same thing. I've never seen so many different varities of mushrooms growing in one place. And none are the common ones you see growing on the ground. I've been just letting mine die down and go back into the bale. Don't know if that's a good thing, or not.
I don't think we have had any explanation for the mushrooms but normally just let them die back, or knock them off if they bother you. I think we have all decided that it is a sign of a good working bale if you get the mushrooms. If you are concerned about the dog, get rid of them. But I would not try to stop them from growing because whatever you do for that might make your bales kill off your plants.