Keith knows his stuff! Id do that.
And Maybe....
After the magic eraser you might want to coat it with a warm blast of urine. Make sure you get all the sides and the bottom. The sterility of the urine on the plastic will create a protective coating like a car's clearcoat. Not to mention it will forever make it "Yours". If you run on empty during the first coat I heard cat urine also works. Repeat process as necessary.