Mike,
Are you sure about the stroking part? The Gemara says that Rabba slid his child on the sides of his donkey, for play. He explained that as long as he’s not sitting on it, the prohibition of using an animal on Shabbos doesn’t apply. Obviously it wasn’t a problem of Muktza, either. The only thing is that in that case, you are not actually moving any part of the animal, while by stroking you are picking up and moving its hair.