If you decide to try a fusible, I would recommend fusible tricot. It is soft and leaves a soft hand to the fabric. Tricot is the stretchy fabric often found in under panties; they add fusible dots to one side.
JoAnn Fabrics sells it in little packets, but I found it much cheaper from an online drapery company -- $2 or $3 a yard, and the fusible is 60" wide. They had it in black and white. I think the company was Black's, but I no longer have the link. Found it by comparison-shopping on the web.
You could buy a packet from JoAnn's and, if you like it, order in bulk from a drapery company. If you use this, hold a steam iron above it for a few seconds to allow the fusible to shrink (you can watch it shrink) before lowering the iron to fuse it to the fabric.
Another option would be to wash the satin first, then starch it heavily (1:1 solution of Sta-Flo liquid laundry starch and water applied with a painter's brush), throw it in the dryer, and then iron it before cutting.
I would hesitate to paper-piece with Satin because it is so slippery, and because I would be afraid of creating runs in the fabric when taking the paper out.