My rabbit also loves to dig and rip up my carpet. We have put large bed sheets down on the floor usually with a few heavy things on top to keep it in place. Once it gets dirty we just wash it and put it back down. Also I learned from someone else on here that rabbits love fleece..so maybe a fleece blanket? I have a flemish giant, and he has free roam in my house when i am home but when i leave he has a gold retriever exercise pen. I bought a king size fleece sheet set and laid it completely flat then set up the pen on top of it-then to keep all his hay and mess in his pen I took about 6 inches of the sheet and ran it up the sides and clipped it with binder clips. It does a very good job keeping the mess contained and keeps my bun from digging or ruining the carpet. It does take some time to set up but once you get a hold of it it gets faster every time. I can post some pictures of Joe’s pen set-up to help show you since its sorta hard to explain

My bun loves rugs and carpets in all the wrong ways, so he had to learn how to run (and make spectacular fast’n’furious styled drifts) on my ceramic floor.
He gets a fleece blanket on the floor when I can supervise. The flooring of his pen is two hardwood boards I nailed together, so he has complete grip when he’s unsupervised and closed off.

About cheap “furniture” that you can lay around: have you tried using cardboard fruit crates? They are stackable so with some creativity you can create interesting towers or mazes. They can be chewed and you just throw ’em to the paper bin when they’re too damaged.