Back to the original problem of your circles curling. What type of fabric are you using? Is it the same front & back or is one much thinner than the other? Did you trim and/or clip the seam allowances before you turned them right side out? Since you will be sewing down the arcs anyway, the curling may not matter. Make 4 of them and loosely stitch them together then tack down the 4 arcs in the center. Does it look ok? If so, continue on. You will love it!
Back to the Faux or Mock question. While I was making my last one, it crossed my mind that the method could also be called Faux Orange Peel or Faux Robbing Peter to Pay Paul.