No, you can't do the bag for a flat price, that would be great, but no.
I can tell you however that having disney do your laundry on the ship is excellent. Everything comes back perfectly folded and pressed, tshirts, shorts, pajamas, everything. We have used it on both of our long cruises now. We take our favorite outfits and let disney freshen them up 1/2 way through the cruise. Definately better than lugging extra clothes, which we never wear anyway, because we like our favorites.