Not sure if this issue is related or not however when I disable the hotkeys I have to play a song manually inside of MusicBee before the media keys on the keyboard work. The functionality of the plugin isn't initialized until a song plays for the first time or something along those lines.
I have also tired your second solution however that prevents the buttons in the overlay from working.
Thank you for taking your time to look at this problem.
Also while trying to figure this out myself I ran across a similar plugin for foobar2k its approach to a similar problem is to passthrough the keyboard messages, not sure if this will be helpful to you at all but here is a link to the plugin source code
https://github.com/Hual/foo_mediacontrol/blob/master/foobar2000/foo_mediacontrol/message_listener.cpp.