Unfortunately the workaround doesn't fully work. It turns out that if I disable the play/pause key for just the MusicBee app, then play/pause works when MusicBee has the focus, but I get the double recognition (play then stop or stop then play, as described above) if another windows has the focus.
But if I disable the media play/pause key for all applications, then I can no longer use the key for other applications, such as VLC media player.
I tried going into MusicBee and setting play/pause to not be global. Then I disabled the play/pause key in the Microsoft Mouse and Keyboard Center, just for the MusicBee app. Now the same thing happens: I can use play/pause within MusicBee, but when another has the focus, I get the double-input problem.
I don't understand why MusicBee can't fix this, if other applications such as MediaMonkey don't have this problem.