I just wanted to add in my newest solution:
Swing by your local hardware store and pick up some .7 mil drop cloth.
Double it up in the hoop, run your outline then zig zag the fabric in before doing the center stitches and finally the satin border. (My fabric is stabilized using heat and seal light hold then laser cut. I also use spray adhesive to position the fabric in the hoop.)
The patch will pop right out of the plastic and, if there's any plastic still showing, you can hit it with a heat gun to melt it.
After that I apply my heat and seal iron on backing and it's good to go. This works for my little pe770 so it should work even better with a proper commercial embroidery machine.