I'm having a hat win!
I've decided to put the grossgrain straight down on the white linen, 3cm apart (that's twice the ribbon's width) and using backstitch. I did 7 last night while watching TV, and have a tiny bit left to do before I start on the other angle. It's not that annoying or time concuming...although I spose I did watch close to 2 hours worth of Gormet farmer and Torchwood.
It's lined with voile, but I found that it was all too stretchy still, so I ironed some light interfacing on it too. We'll see how that works out when I start to pleat it in.
I now own some thinner red ribbon in silk, and I'm yet to test the pinchy theory. Again, if it just isn't working, I'll still have some tasty red ribbon to put on coifs. And aprons.
Starting to wich I'd picked a linen scrap that can be ironed flat - this linen is dumb. I've never been able to get it smooth. Oh well, too late now.