You can't iron Minky, because it will melt. But, you can use a washable glue stick or Roxanne's Glue-baste-it. Maybe you could stitch it down with a basting stitch before you finish the raw edge?
I imagine there might be an embroidery machine design for Cookie Monster, but don't know if you have an embroidery machine. I don't have one, but I've seen some really cute things done with minky on an embroidery machine.