One idea I had is if you could drill a spot weld hole (or two) through the sheet metal directly in front of the nut, you'd be able to spot weld the nut in place. This would be an easy quick grind and paint once finished as the affected area would be relatively small. The hinge plate would very likely cover up the burnt area if you're quick and handy with a mig.
Chase the threads with a tap might be all you need to do to "fix" the nut enough to get a tight hold. If not, I'd try a fine thread bolt or 12mm bolt (after tap) which is similar size but far different thread pitch which may work without getting a bigger hole.
Once finished by any means, please share with us how you did it?
Last edited by FootStomper; 12/12/2016 8:03 PM.