Our guinea pig started in a cardboard box. At that time, the household was 4 cats and 2 large dogs. Patches stayed in cardboard boxes for two years. It was easier to throw out the box than it was to clean the cage and my daughter was not too good at doing either.
During the cardboard box era, the cats would stick their paws in the box to try to bat Patches. One cat discovered that guinea pigs have very sharp teeth ($200 vet bill for the cat's paw). None ever tried to jump into the box with him.
So, if you want the open cage, I'd suggest you try the cardboard box first and see what happens. Line the bottom with the puppy training pads, add whatever you use for normal fill on top of it, drape a towel over one end so she has a hidey place and then you can see how the cats behave.
