I do machine applique, and instead of using stabilizer I heavily starch the background fabric before cutting it into squares. Basically I mix a 1:1 solution of STa-Flo laundry starch and water, "paint" it on the fabric until the fabric is saturated, toss the fabric in the dryer, then iron with steam. Most of what I do is invisible machine applique, but I have successfully used this technique with fused applique and a satin stitch.
If you have already cut the background squares, try using Stitch and Ditch underneath the background fabric. I got mine on Amazon. It is very lightweight, so is easy to remove afterwards.