You could use Vetbed. It's a fabric bedding actually designed for puppies and older, incontinent dogs, and is used by vets in their cages.
It's designed so that if it gets wet, the urine drains through, so the dog isn't walking or laying in its own wee. You can put something like newspaper underneath to absorb the wee that does drain through.
Vetbed is also machine washable so once it gets too wet or smelly, you can wash it to kill any germs and get rid of the smell.
You could also use a littertray if your bun will use it. Pelleted cat litter is good for that - though make sure it's a wood only one, not one with silica or clay in it as this can harm your bun if he eats it.
There are also cardboard/paper substrates which may be less likely to get stuck in his fur, and you could use these all over the cage or just in a littertray if he uses one. Carefresh and Megazorb are one type (both very similar but Megazorb comes in bulk whereas Carefresh doesn't). Ecopetbed/Easibed is a chopped corrogated cardboard bedding designed originally for horses with respiratory problems, but works well for small animals too, and doesn't cling to the fur. The only problem is finding a supplier - though you can order them online you pay up to 3 times as much as the actual product for the postage costs!
It might be worth trying a horse supplies/farm shop near you to see if they do any paperbased horse substrates that you could use.
And if he is littertrained, you don't necessarily need a substrate all over his cage/hutch, just in the littertray. If he has a hutch and you're worried about accidents, you could line the bottom with an offcut of lino to prevent any wee from absorbing into the wood.