Wow, so sorry this has happened to you! Is it possible to cover the fleece backing with a tightly woven (maybe batik?) fabric? I was thinking maybe hand stitching it all along the edge of the binding on the back. That would contain the lint that comes off the fleece, if that is where the lint is coming from. You might be able to take some tacking stitches in it which don't go through to the front, here and there, just so the new backing is not loose all over.