There is a thread that also deals with the cooperation of MediaMonkey and MP3tag.
Mp3tag corrupts every mp3 file after editing the tag .
In short: don't use MM.
Also: it has been noted by other users in the forum, that sometimes MP3tag cannot access files for writing.
This seems to orginate in the Explorer that also reads (and blocks) the files while they are being updated. This is not quite the same as your approach but you tell us that you see the updates in WE. It may be a better idea to let WE point to a different folder and not to use the WE context menu on a file to open it in MP3tag but use the navigation from within MP3tag to open a folder.
Don't know if this helps