You don’t need to use cage bedding. 
In fact, it will help with litter training if you don’t have it, just have litter in the litter box. Your bunny has obviously shown his preferance of lying on the bare base. That is pretty typical of most rabbits. You could try something like a fleece blanket but when litter training, anything soft and absorbant may become the new toilet. So it that doesn’t work out, maybe try introduce again when he’s a bit older. Rabbits prefer the cooler weather, so they do like to lie on things like tile when they weather is warm. I think I’ve seen suggestions of using an outdoor mat to line a cage base, but turning it upside down. Perhaps something like that would work for Lychee?
It’s also typical of a rabbit to use the litter box as a bed. You could add in just an empty litter box or cardboard box and see if he takes to using that instead. This sometimes works with my girl who often lays in the litter box. I give her a shallow cardboard box and she spend her time in that instead but they never last long, as she deconstructs it. Easy and cheap to replace at least!
ETA: Bonus pics. She’s the white bunny and because she likes to sleep there, he likes to be where she is – if he can fit.
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