I have had two mirrors damaged by accident and have been through this exercise without success. The plastic of the retaining ball and socket gismo appears to get hard over time and requires a lot of pressure to release.
I had a mirror cut by a local glass supply outfit and glued it in, only to be told that they could have fitted it to the frame by softening the edge using a heat gun. Therefore it should be possible to remove a mirror the same way to get at the screw behind. Live and learn, as I hope we all are as a result of this thread.
P.S. If anyone is successful in this, details should go in the how too section, as it is a perplexing issue.