I would look for 1930 reproduction fabric yardage in a turquoise small flower or other tiny motif. I would sew the new fabric border right over the whole thin border on the outside. Just move the stitching line a little more into the white so that you have a stronger stitching line. I would run a little strip of batting join tape under the split section to reattach it and then hand quilt the whole quilt again. I would bind it with the same yardage. This way you haven't really changed the quilt but persevered it as much as possible.