We had a similar problem. Sascha (husky mix) got deathly ill. After many (expensive) tests, the vet found that Sascha has pancreatitis. Sascha had been snacking from the cat box (kitty rocca ) and had a massive infection. We tried a covered cat box, but the kitties were afraid of it. So, we solved it by moving the cat box into the laundry/utility room. Then we cut a hole in the door and installed a cat flap. Kitties go in and out. Sascha stays out. Everyone's happy and Sascha is healthy. Do you have a room/garage that would work for you?