As the title says, there is no mapping for MusicBrainz Composer ID.
I tried to add a user defined mapping from "TXXX:MusicBrainz Composer Id" to MUSICBRAINZ_COMPOSERID. That works on import, but writes the frame description back as "TXXX:MUSICBRAINZ COMPOSER ID" in upper case, which is then no longer recognized by MusicBrainz Picard.
(The latter can be mitigated by enabling "Reuse unmodified ID3v2 frames" if the file has originally been tagged with Picard, but at present it is not possible to add a missing Composer ID directly from Mp3tag.)
Could you try an action of the type "Case conversion" for _FIELDNAME and the setting "Mixed case" and see if you get a better result?
I can not reproduce this behaviour.
If I enter a new field manually in Mp3tag, I can see it in MB Picard:
Added some text "from Picard" in MB Picard and saved into my test track, I can see it back in Mp3tag as expected:
I'll add this with the next release. Thanks for pointing!
Thanks for the suggestion. That lets me change the field contents (i.e. "7a8dbe84-f4c0-4457-bfa3-edced1f8cde0" ↔ "7A8DBE84-F4C0-4457-BFA3-EDCED1F8CDE0"), not how the frame name is stored.
Basically this is the same as Writing back MusicBrainz IDs in all Uppercase? , just with another frame.
Your screenshot shows the exact problem 
It should show up in Picard as "Composer MBID", not as "MUSICBRAINZ COMPOSER ID".
How do you set this "Composer MBID" manually in MB Picard?
AFAIK there is no such field name in "Add new Tag" in MB Picard.
Do you use any addon for this purpose?
(Or do you use the brand new released MB Picard v3.0.1?)
Sort of ... I'm using 3.0 - I didn't even know that 3.0.1 is already out 
It might indeed be new in version 3. It's right there with the other musicbrainz_* tags:
I've added the mapping with Mp3tag v3.37-beta.1.
