Our cats won't use litter boxes with lids either. They are Maine Coons and really long, tall cats so it is uncomfortable for them to try and turn around. Hubby made a wooden box out of ply wood and 2x4s. It is around 4' long, 2'wide and 2.5' tall. The lid is made so the 2x4 frame underneath the lid fits into the top of the box, so it is secure. He covered the entire thing inside and out with carpet to match the cat tree. The carpet absorbs any odors and the carpet on the floor keeps litter inside the box. We recover it every year or so. Two large litter trays fit in it, since my cats like different types of litter and won't both use the same one. The box is in front of our dining room window. They use the carpet covering as a scratching post, the cats and dogs lie on top of it to look out the window, and with the hole in the end, we slide the cat tree up to the end of it. The cats can get in, the dogs can't get to the opening and everyone is happy. Joe made a shelf that fits on the top perch of the cat tree and that is where we feed the cats. My dogs are so heavy, if they try to climb on the tree to get the food, it tips and they get off before it falls. Works well. Good luck.
Kathy