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Your wife was right. Use a eyeglass screwdriver to apply a little pressure downward on the screw through the rectangular space. Then screw off with a screwdriver. You'll get a little bit of space, slip an exacto knife between the screw head and the case. Then finish screwing and you can get it out.
I have to add to the dogpile. The head on the screw seems to need some sort of tool that I've never seen, a torx 15 or so with a hollow point. I ended up using a long nose pliers to grip and screw it in place. Thankfully I have power to the doorbell and won't have to remove the battery to charge it.