Don't you just love it when a project talks to you? I had one yell at me once. It was kind of annoying, lol.
As long as the satin stitch covers both edges, you should be fine with the plan. The tunneling (or pulling) occurs when there is a difference in thickness, which you won't have except for the outer edge. When I run into that, I make sure that I put the entire satin stitch along the edge of the applique.
Don't know much about the wash-away stabilizer. One alternative for using regular iron-on interfacing is to draw the shape on the interfacing and then CUT OUT the center portion, only leaving a 1/4-1/2" wide piece that actually gets fused. If you do that, you really wouldn't need any stabilizer at all AND you would not have any stiffness from the interfacing, except for the bit along the edge.
Hope this helps.