Can you remove the quilting from the center of this block and then rip out the stitching holding the white to the outer dark fabric frame? Then remake the inner part. Instead of using 2 pieces RST, use a piece of lightweight interfacing as the backing. Sew all around, slit the interfacing, turn right side out and press. Now you have a finished edge piece that you can match up to the original seams. I would blind hand stitch the new piece to the outer frame, then re quilt the center motif.
Let us know how it goes.