I would definitely give the fleece a life after death. If you don't want it to show, go ahead and use it as batting and then use muslin, a bedsheet or whatever for the backing. Since the fleece is not going to shrink, I would probably pre-wash all the fabrics for the top and the backing.
When quilting, you will want to start in the middle & work toward the corners. Fleece is stretchy so you'll want to pin or spray baste. Avoid tugging on it. You can tie it by duct taping it to the floor, wall or table. Again, start at the center & work your way out to the corners.