Anything else I could try to reach a color tag mapping between PM and CO?
Warning: Long post, and: you may have to experiment just a little bit more
The Capture settings shown in attached image are from my Capture One 20 (or v13 if you like). It may look different these days with Capture One 23 or v16, but the point is to not prefer one metadata set over the other and not do auto sync. Experiment with the manual sync functions instead. Capture One has two of them in the Metadata tool (circled "i", then "..."). The first is "Reload Metadata" to read metadata saved by PM. The second is "Sync Metadata" to write metadata to the sidecar (.xmp) file for PM to read.
Set PM to update both embedded and sidecar (as I understand you already do). I attach an image showing part of my PM-dialog for those settings. Pick one image for experiments and set the color label in PM. Switch to Capture One, select the same image file and do Reload Metadata. Capture One should instantly reflect the color label set by PM. Do the same for all PM's color labels, one by one. Set in PM, switch to Capture One and Reload Metadata. Make a note of color label set in PM and the color Reload Metadata brings into Capture One. After each step in this first color map process, Capture One deals with three color labels: the embedded one, the sidecar one and the one in Capture One's database.
Now, turn the table. Change to a different color label in Capture One and do Sync Metadata. Switch to PM and make a note of color label set in Capture One and the color shown by PM. Repeat for all color labels in Capture One, change color label in Capture One, Sync Metadata, switch to PM and make a note of color set and color seen. After each step in this last process, PM will see two color labels: The embedded label last written by PM itself, the sidecar label last written ("Sync'ed") by Capture One. Sync means copy from Capture One' database to sidecar file. (PM+ may also have a third color label in its database).
Color label set at multiple locations (embedded, sidecar, database) may cause some confusion. I have experienced and also seen many posts where Auto sync causes Capture One to briefly (a few tenths of a second) display metadata set by an external application and then reverting to the previous value. I think of it as a race condition where value read from one location is quickly replaced (sorry, Sync'ed) from Capture One's database. It may be so brief that it escapes the user's attention, leading to claims like "PM does not change metadata". This is why I recommended to make PM write metadata to the image file (embed) and also to the sidecar to make sure Capture One reads the value PM wrote.
Back in the day when I first mapped out the set-seen colors as outlined above, PM did not provide a preset for Capture One. I continued using my own as shown in the attached dialog. The color order (number 1-8 in the PM dialog) turned out to be important back then, it may no longer be. You will notice that PM's black color label was not recognized by Capture One back then, and i think I was not able to make PM pick up a color label set to "None" in Capture One.
Once you make colors match, you can try out combinations where PM writes to sidecar only and Capture One prefers one set of metadata over the other, and finally auto sync if you need it. You may or may not find a combination you like.
I ended up with a workflow where I do all metadata work in PM (now PM+) and did a manual Metadata reload if I wanted the metadata in Capture One. I probably should mention that I abandoned Capture One a couple of years ago and should probably not post Capture One related stuff anymore, though I hope this may help a fellow PM-user.
