Since the quilt is already quilted I would go ahead and bind it, rather than do a second back and birth it. You are asking for trouble IMHO to do a second back and then try birthing. You will have a backing that isn't connected to the quilt in any way, except the outside edge. I think you may end up with some unsightly ripples and pleats and after you wash it the first time it won't look very nice at all. It is really easy to bind a round quilt. You just have to cut your binding on the bias. Use a walking foot and take your time. I think you will be much happier with the end result. Here is a link to one I did.
http://www.quiltingboard.com/picture...e-t191197.html