I think the cause of this problem is satin stitching that is very tight. I have noticed when I do satin stitching that is very close together, the needle holes in the applique are so close together that inevitably a lot of the applique fibers are going to be damaged. It's almost as if the satin stitching cuts the fabric. That is why the applique ends up lifting right at the stitching line of the satin stitch. The fibers under the satin stitching are still there, but they have separated from the body of the applique.
The best solution, I think, is to change to a different type of stitch, such as the blind hem. Or, widen and lengthen the satin stitch so the stitches are not so close together. Another alternative is to do two rows of stitching, both with lengthened stitches. The first round is not as wide; the second round with a widened stitch covers up the first round. That way at least the needle holes are not right next to each other.
A different needle might help also. For example, a ballpoint needle is supposed to "push" the fibers apart where as a sharp needle will pierce the fibers. I think it is the repeated piercing of the fibers so close together that is weakening the fabric.
Possibly a different fusible would help also -- one that is more permanent. Regular Steam-a-Seam I noticed stays adhered very well, but it is pretty stiff.