I do not use a stabilizer... I either fuse or use a tiny amount of Elmers glue (and heat set with the iron) to hold the pieces in place. The double thickness of fabric (or more thicknesses) is plenty of stabilizer for me.
I think the freezer paper would be hard to tear away and it could distort your stitches. Blanket stitches are not as stable as a satin stitch.
Try blanket stitching on some scraps and see if you think you need a stabilizer. :D:D:D