I haven't read all of the posts, but QUILT POLICE, COVER YOUR EARS!: I would cut a straight border and just applique the curves fat onto the border, either by some type of appropriate decorative stitch (zigzag, satin stitch, etc) or by turning the edge under and doing a blind handstitch. Then I would add the pieced border to that. I know it's unconventional, but it would give a straight edge and the character of the rings would remain. Good luck in whatever you choose.