I dropped the contents of the x64 folder into the main Musicbee directory & Musicbee went crazy with errors when I tried to select asio. So then, I dropped all the contents of the zip into the main directory of Musicbee leaving the x64 as a subdirectory. I was able to select asio, but The new bassasio.dll did not fix the issue.
Since MusicBee is a 32-bit application, you can't use 64-bit plugins with it, I just assumed you were aware of that when I pointed you to the bassasio.dll package or I would have gone into more detail; BASS libraries commonly contain versions for both 64-bit and 32-bit, with the 64-bit always being in its own subfolder. From the zip package, the only included component you should have in your MusicBee directory is the 32-bit bassasio.dll, be sure to delete the x64 subdirectory in case it may cause future issues. The two "bassasio.h" and "bassasio.lib" files are not needed.
At any rate, I'm glad to hear that wasapi has solved your problem! Exclusive mode will bypass the Windows mixer and often provide better sound, but yes you would need to use shared mode or DirectSound if you want to be able to use other audio applications while MusicBee is playing. Note that pressing "Stop" when MusicBee is using wasapi-exclusive will free up the audio device without having to close MusicBee.