If it looks like the top piecing will hold, you could just buy a new piece of flannel big enough for the whole back, and pretend the original backing and quilting aren't there. Quilt the piece, bind it in the usual way, and be done. You will have preserved the original quilter's work (the top) and won't spend a lot of time with unsewing, which is bound to result in frayed fabric and lots of frustration. The recipient will be thrilled to have it done and won't notice that one side is thicker than the other. (If that bothers you, you could first lay another piece of flannel over that part of the back.)