I am trying to fish a cable up from an outlet/switch box, over a door, and back down to a box outside. What is the best method to fish the cable?
Accept the need to do drywall repair.
Knock some access holes. Unless you are going into an accessible (attic, say) space to go "over a door" you need to drill holes in studs - and if there happen to be holes there now, you still are not going to be able to fish through them without some access holes, IMPE.