Yikes...

Being recessed like that is a nightmare.. You know what I'd probably do with that? Drill it out.. Do you have a drill? If you do, do you happen to have metal drill bits - bits that will cut holes in metal? If not, a regular bit might do the trick, but might ruin the drill bit.. you need a drill bit about the diameter of the rivet, then you just drill right through the rivet head.. being recessed like this one will help keep the drill head in place.. When you go to drill, you only need to get through the head - the pry a screwdriver under the battery door on the opposite side and hopefully you can raise the door right over the remnants of the rivet - when the door is open, use pliers or something to remove the rest of the rivet.. then use a tad larger sized screw to screw the door back in place..

If you just happen to have a drill press, that would be the absolute best way to go.. I got one for Christmas, but haven't used it yet..

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