My Beenius stopped creating a separate log file since version 1.3.4. I use MusicBee 3.5.8606 Portable and Beenius 1.3.6.
The log was broken indeed if the plugin is used with Museexmatch or another one that is using NLog. Since all the plugins are running in the same scope, NLog is global as well and is expected to distinguish log targets between different classes. I've attempted to fix this multiple times, this one is final (I hope).
Have you confirmed that any changes you made to the defaults inside the beenius.conf file are still present? Please post a shot of its contents--it's possible that you may have accidently damaged the syntax in a way that makes it not visible to the dll.
The plugin .dll was working from any Plugins folder, but .conf file was hardcoded to "./Plugins/beenius.conf" since day one. This is expected to be in Program Files and it's not writable. Now it's moved to %APPDATA% and will remain there.
As for the portable version, I guess we've discussed this a certain time ago and by then I've suggested to ask Steven to remap %APPDATA% to something in the portable folder. Don't know how did it end up being, but you always can just manually create MusicBee/Plugins in AppData/Roaming and it should work.