How to move a garage opener switch and minimize damage to sheetrock
Because it is drywall, you can do pretty much as you suggest:


Enlarge the existing hole enough so that you can drill a hole through the stud
Cut a hole for the switch above the light switch box
Take the door switch off the cable and feed the cable through the stud
Use a coathanger/hook/other tool to catch the cable and pull it through the new hole
Reattach the switch and install into the wall
Plaster and smooth old hole
Paint as necessary


Job done.