Did you try using plastic polish? This is something that I have tried and found useful. I suppose any commercial plastic polish would do the trick. First, clean the lid thoroughly with a soap solution and dry it completely with any lint-free cloth. Then, take a small amount of the polish on a cloth and apply it on the lid slowly and thoroughly. Rub everywhere in circular motion and make sure that the polish is completely absorbed all across the surface. Then use a clean cloth to wipe the lid again. This should get rid of the scratches almost immediately. You can also repeat the process until you see the scratches fully gone.
Hope this technique works for you! Do let us know the results.