Ok so I've been putting the hay on top of the litter as so many others do but isn't that unsanitary? And is touching the poop with your bare hands bad ?
Bunnies eat their poo(special soft poo's usually at night so we can't watch lol) so I don't see an issue with putting hay in the litter boxes, I try too put the hay in the part of the box they don't pee in. So they can sit in the corner do their business and munch away on a snack. I look at the poo as pretty much just processed hay. But like every pet/animal wash your hands after handling them and before you eat.
Have you tried the covered litter? You can put a grate on top of the litter (the stuff for plastic canvas works too) Runestonez has a fabulous DIY for this and it's outstanding. The hay stays on top of the grate & they can munch it even if it's all over. I have a wire toothbrush holder that I use for hay and put that in the litter