I am less than impressed with the auto kitty bx -- it seems to get stuck all the time - I've spent more time trying to get it to work, making sure I didn't add too much litter, etc. than it takes time to clean out a traditional litter box.
Does your cat(s) need a litter box on every floor of the house - wouldn't it just be easier for the cats to go up/down the stairs then you don't have to worry about the yorkie in the box? Just an idea.
I just bought the most wonderful litter product last week at the Great Iowa Pet Expo - It's called LitterMate and it really cuts down on traditional cat box concerns - here's a link to their website - maybe this would help you - definitely a money saver!
www.littermate.com