I used to work at a fabric shop (err- money pit for employee) and always wrapped ribbon, elastic, etc. around thumb and baby finger in figure eight design. No wrinkles and creases to touch up with an iron.
Try this with your binding. Place the bundle on your lap and you'll find it works smoothly and doesn't fall off. Leftover strips (bundles) stored in see through container for scrap-type bindings. No pins, paper or rubberbands touching the fabric....:-)