This sounds like a good time to experiment with Elmer's School Glue!
I think I would put a thin coat on the back side of the Minky to stabilize it so that it can't stretch. Then, I'd glue the applique piece directly onto the Minky pile to prevent it from shifting while machine stitching it down.
Once the sample's glue has dried it will be time to test out the technique! I don't know if it will work but I can usually do anything if there isn't slipping, stretching, and sliding going on.