For this type of applique, instead of using a stabilizer, I heavily starch the background fabric before cutting. This makes the background fabric stiff enough to prevent "tunneling" and other distortion from the applique.
My technique for starching yardage is to mix a 1:1 solution of Sta-Flo Liquid Laundry Starch and water, "paint" the solution on using a large wall painting brush until the fabric is saturated (I do this on my kitchen island), toss in dryer, then iron with steam.