You could juse natrium hydroxide. Be careful though, it burns you and will discolour and damage the wood somewhat. Rinse well and let it stand for a while, the hydroxide turns into salts when rinsed enough and exposed to air.
I'd leave the paint though. Most cages here are painted. Just put some metal on edges (like beams in the corners) to avoid gnawing on those. Provide hay and/or chewing sticks as alternative to the cage. Rabbits don't really gnaw on flat surfaces, at least I have yet to see it in a case where they didn't start at a beam or edge and continued through the floor.