Download and enable the Classical Extras plugin. Give it a try. It seems to have a lot of capability to put all the classical music participants into Artist and Composer tags.
Forum discussion at Classical Extras plugin . I don’t know it well yet. I am just beginning to start using it.
And, if you don’t know beets, you might want to give it a look. You might find its tagline fits: “the media library management system for obsessive-compulsive music geeks.” Eventually, it might replace Picard for obsessive tagging.