stitch 1/4" around the edge of your outmost border.
make a solution of laundry detergent and water in a nice deep container, sink, or bathtub. put the quilt into the solution. swish it a little and let it soak for a bit.
your quilt won't fall apart from such gentle handling. if the batting disintegrates, it will only happy in the 1/4" along the edge. that won't injure the quilt and might even make the binding easier to apply and turn.
i don't think you have much to worry about. if it's made for kids and says it's washable, it will almost certainly come right out.