Maybe you could color the dough white? I'm not quite sure how good this will work, you will probably only get an off-white at best. But in contrast to the red and blue it would look whiter in the end, I suppose.
Or try painting the surface of the cookies with white. That works pretty well, if you don't mind that it is not really a thick coat and the structure of the cookie still shows. I mix Americolor Bright White with a few drops of Vodka for this purpose and paint it on with a large brush. It doesn't take long, dries pretty quick, and you can pipe the icing on top of it.