I had the same worries but honestly having no bedding is the best solution for me personally because my buns would pee on anything soft! So I only use carefresh in their litter boxes and the rest is bare. The bunnies are happy to lay just about anywhere, sometimes they’ll flop out on their cage base (plastic), sometimes the carpet, sometimes their play mat and if they really want to be a pain in the butt they will push the carefresh aside in their litter box and flop out there. I thought about introducing fleece bedding after they’ve been neutered (currently they are great at peeing in litter box and nowhere else but I want to be sure this is ingrained in them – saying that though, I’ve put a fleece jumper all folded up in Buttercup’s cage after bringing her home from being spayed and aside from peeing on it the first day, she hasn’t peed on it since!)

In nature rabbits don’t line their burrows with anything except the mother’s when they have newborn babies. They wrap their babies in their own fur. So it’s natural for buns to not have bedding where they sleep.
They do however have a need to dig, so it’s good to offer that possibility. You can make a dig box or let him have a cheap fleece blanket to dig in. They do like to pee on soft things, as Sarah says, so beware of that. Don’t give him sth you plan to use yourself later, because he’s likely to destroy it over time.

My bun does exactly the same. I use hay for bedding and she just kicks it and sleeps on the bare cage.

My rabbit has pine pellet litter in his litter box, but his crate lining is a braided rug welcome mat and old curtains from the thrift store. They are large enough they have to be folded to line the cage and burrowing around the layers of folded fabric seems to satisfy his digging need. If it’s a huge curtain I’ll even cut it and use the other half when the first half gets too worn out. He also loves to rip up the braids on the braided welcome mat! It gets replaced every few months and is super cheap from goodwill/valley thrift etc. Before the rug/curtain getup his only issue with a bare crate was traction!

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Fleece is a good option – as members said above, one downside is they may pee on it.

My rabbits currently have a pet store cage WITH free range of a room (Don’t worry, it exceeds the minimum by quite a bit.) In the cage I use Carefresh, but everywhere else they’re foam exercise mats covered with fleece. They can’t chew the mats either, by the way. (Once bonded they will have a x pen with fleece or free range of whole house.)

I also just recently stopped using bedding. They were kicking it out of the way and I was going through a $20 bag of bedding a week (between two connected rabbit cages and four connected hamster cages), so I just… stopped. So far, it’s going good! It’s been a few days, and it seems to be going okay. I’ve only seen pee and poop in the litterbox!