I don't use stabilizers for machine applique. Instead, I heavily starch the background fabric (1:1 solution of Sta-Flo and water). This stabilizes the background fabric sufficiently so I don't need any other stabilizer. I have used this successfully with all types of stitches, including satin stitch. No tunneling or distortion.
I use a fast way to starch. After mixing up the Sta-Flo and water, I "paint" the solution onto yardage using a large wall painting brush until the fabric is saturated. I wait a couple of minutes to make sure the starch has had time to penetrate the fibers, then toss in my dryer. Ironing with steam re-activates the starch so I have a very stable fabric to work with.
Starch washes out completely, so there is no paper to fiddle with.