When doing satin stitch, you always need to stabilize the background fabric; otherwise you get tunneling.
My unconventional method is to heavily starch the background pieces before cutting. I use a 1:1 solution of Sta-Flo liquid laundry starch and water, "paint" this on with a large wall-painting brush until the fabric is saturated, toss in the dryer, and iron with steam. When I cut the background blocks, I always cut them about an inch larger and trim them to correct size after the satin stitching is done (probably overkill, but I don't want my block to end up short due to take-up from the satin stitching).
There are teflon-like applique sheets you can buy to protect your ironing board and iron when fusing. The teflon sheets used for ovens work too. What I am beginning to gravitate to is simply two sheets of parchment paper (used in cooking, sold near the freezer paper in grocery stores). It protects just as well as the teflon sheets but seems to prevent any "shine" from developing on the top of an applique from the fusible.