I use a grate over the litter box. This prevents my rabbits from sitting in the litter and keeps the litter out of the fur. Hay is in hay racks that sit in the box and are a bit above it. This way most of the hay mess is contained to the litter box which makes it easy to clean. I use fleece in the rest of the cage, so there is nothing to stick to the furry buns.
You can try trimming the bum area and the feet so there is less wool for stuff to get stuck in. Also keeping the tummy short helps. If you are going to show him, then you could keep some areas shorter when you are not showing and grow them out when you plan to attend a show.
Keep him well groomed. Blow out the coat several times a week as this will get the hay and other stuff out of the coat and help prevent mats from forming.
You really just need to keep the cage clean. Set up the cage so you can easily sweep up the hay and poops every day. I use foam puzzle mats in Lillian's cage and that makes cleaning much easier, I just need to sweep the vacuum the little bits and wipe it down.
