For something like that, I starch the background fabric very heavily before cutting. This is how I do it (fast and easy).
Mix a 1:1 solution of Sta-Flo laundry starch and water. (Sta-Flo is sold in a blue quart bottle in the laundry section of most grocery stores.) Lay the fabric out on my kitchen island and "paint" the starch onto the fabric using a large wall painting brush. I saturate the fabric, then toss it in the dryer. Finally, I iron it with steam. This leaves the background fabric about as stiff as cardstock and very stable; there is no need to use a stabilizer under it.
I also starch the applique fabric, although not as heavily.
I prefer using the very tiny, short applique pins to secure the applique to the background. You need at least 3 pins to prevent shifting, but these can be an inch or so away from the edge.
Starching the background fabric heavily prevents "tunneling" of the satin stitch and eliminates the need for a stabilizer underneath. I normally don't use spray starch because I have a lot of trouble controllling over-spray, plus I tend to have my iron too hot and have both gunked up my iron and scorched the spray starch. None of that happens when I use the Sta-Flo method above.