I've done what Eva suggests many times. I'm always afraid I'll screw up my scarf material! lol
A tear away backing would be a good idea. Also use Solvie as your stabilizer. You lay the Solvie right on top in the hoop, then after the design is done sewing, you wet with a water bottle, which dissolves, and tear away the extra that is not around the embroidered design.
It sounds like Eva has the answer - but I don't think I would do it any other way - either. *4U
Will pucker.
What about stitching on a foundation material and use the ready made stitchout as an applique on the scarf? You could use this melting applique helper, which will prevent the material to stretch while appliqueing.
I don't know but send this TTT for other Cuties to help you. :)