I have a storage closet room (about the size of my bathroom) that I turned into the cat litter room. I installed a kitty door so the cats can come and go through the door. For some bizarre reason, Bitsy wouldn't go through the flap although she does fit (she's smaller than the cats) and that's worked for me.
You can try top entry litter boxes:
Amazon.com: Clevercat Top Entry Litter Box:...
or a product called S.E.P (Stop Eating Poop) by Solid Gold. You sprinkle it on the CAT's food so when it comes out the other end, it won't taste good to your puppy. Only problem with going that route is that cats can be such finicky eaters and might not want to eat their food sprinkled with the stuff...