I have told On1 Photo Raw to "Read Metadata from Photo" but this does not do the trick. Is there anything I should be doing in PM to make sure the metadata is really embedded?
If you're not getting an error when setting the color class in Photo Mechanic, then it's embedding the metadata (since you said it's a JPEG). There's ways you could doublecheck this (exiftool, for example, or just send us a file), but ultimately I would guess that there is some way to force On1 to re-read the metadata in your file after the file has been modified. Whether that can be done automatically (like Photo Mechanic is doing) or you have to manually trigger it, I cannot say, as I've never used the program and all of the previous threads on this issue seem to be people who had trouble going the other direction (they weren't seeing changes made by ON1 in Photo Mechanic). I'd probably recommend sending a file that you've rated in Photo Mechanic to ON1's support and ask them what you need to do to get their software to read metadata that has been changed by another piece of software.
Color labels can be slightly more complicated, the set of labels has to match in both programs. Once you figure how to get their software to read your files' metadata after Photo Mechanic updates it, I would see if that allows both programs to see the color labels the other software changed, and if not then we can see what settings might need to be changed in Photo Mechanic.