Well you already know my vote is for cross hatch as that was my recommendation on your original thread. Warms my heart to see you going for it!
If you are planning on showing the quilt the outline stitching will affect the judging. But if this is just for your enjoyment it isn't neccessary but I really think the finished product will look better if you do, especially on the back as Old Rosebud pointed out. Regarding your SITD around each square, aesthetically it won't make much difference however it will be a great stabilizer if you first ditch stich your blocks, then go back and outline the applique, then go back and cross hatch. But you could also get away with outline then crosshatch. Harriett Hargrave's book "Heirloom Machine Quilting" and Diane Gaudinski's book "Guide to Machine quilting" both emphasize the importance of stabilizing stitching in DSM quilting.
another option is to use your applique as a "highway for travelling" in your cross hatch. I'm not sure if you are using your walking foot or not. But I am wondering if you could drop your feed dogs for just the 1" of travel space along the applique edge to go to your next line of crosshatch. I know on LA that is how I do it and no starts and stops. Not sure if it is feasible on domestic. The aforementioned queens of DSM quilting would say ditch the walking foot and FMQ your crosshatch lines. I see you do have them marked so you may want to give that a shot and see how it works out for you.