Not all bunnies object to flooring, especially when it’s hot, so I would take her out onto floor like you have under the carpet and see how she behaves first. Some bunnies love cool marble tiles in warm weather. My bunny won’t set foot on flooring, but she’s funny in several ways.
I’ve never used the puzzle pieces because despite what others have said, I am too worried about bunny chewing. What I tried most recently, and it might work for an area in your room, are flat seagrass mattes. They are smooth and low, and I tied them together to make an xpen floor. Sure enough, Sammy started eating them at the edges, but they are completely healthy for her so it’s not bad to ingest.
I have also used very low pile area rugs from Target (Sarita found hers in the past at Walmart) that are about $20. I can lay them down for traction, or roll them up. It’s a 4ft. x 5ft. one, so easy to handle. Perhaps you could get one of those, roll it out for her to play and run, then tuck it under your bed to expose cool floor for yourself. If your bunny gets used to the floor, she may be OK running on both, or just floor eventually.