In the "olden days", quilters would encase wool batts in cheesecloth to prevent bearding. I think in your case, for future quilts, I would simply provide an extra layer of fabric between the dark backing and the wool. With machine quilting, I don't think cheesecloth would be enough. I would probably shop for thin muslin and -- even though I don't normally pre-shrink I would pre-shrink thin muslin -- spray baste a layer of muslin to the backing before layering. This should be enough to prevent bearding on the back.
Certain types of cotton batting are the closest thing to wool batting that I can think of. Bamboo and silk would both be much more limp/soft than wool.