I'm not sure I can help, but I would assume that it would bleed if washed. I never recommend dry cleaning, since the chemicals are nasty. Since you're using organic fabric, I ssume you want to avoid that. I'm a huge fan of organic, but at the same time, the reason we use synthetics, including newer dyes, is because, while they can be more harmful to the environment, they sometimes "work" better. I would think about the use of this quilt, and decide if you want to continue, unable to wash it, or if you want to rip out stitches and redo it with a different backing.
Good luck!