I have used valve lapping compound on the tip of my screwdriver in the past to give much more grip while removing a damaged screw.
I found this tool online a bit pricey but it may be the ticket or maybe similar ones might be cheaper. I did see something like it at a local hardware store but I can reamer what the price was.
At last resort you can remember drill the head off of it then once the panel is removed work with the mess without damaging the panel. If you try this keep the drill slow and use a sharp bit to keep the heat down most door panels don't like heat.