I think you could do it all with fusible. I would use Misty Fuse because the applique will remain soft. All you need to do to secure the thin lines is to machine straight stitch through them (with matching color thread), which is achievable even if the applique is only 1/16th of an inch thick in place. The edges may eventually lift from washing, but this is simply called raw edge applique. In the thicker areas I would straight stitch along all of the edges.