This is the current paragraph in the Musical Theater style guideline:
Track Artist Credits should be as listed on the release. In most cases this will be the composer. Again, in some cases, the credited artist may not be suitable for a separate artist (e.g. "Full cast", "Ensemble", "Nun's Chorus"). In such cases, use the Special Purpose Artist [theatre], credited as it appears on the release. If there are no credited per-track artists, use the release artist, or the track's composer if there is no clear release artist.

We need to know what to do if the tracklist shows the name of the character (not actor) next to the song title. This is fairly standard practice with theatrical playbills, and the format is imitated on the cover art by a lot of releases. The cast credits are usually elsewhere on the release. There is some confusion whether listing the character name is sufficient to give the track artist credit to the actor performing that role, or if the name is there just to help the audience know what character is singing so they can follow the theatrical storyline. (For operas, the composer is always used for track artist.)

I think our three choices in this situation are:

• Use the composer for the track artist credit.
• Use the name of the actor as the track artist credit. (This will will require editors to map the characters names to their respective actors.)
• Use the actor for the artist credit, but change "artist as credited on this track" to the character name.

Some examples:
http://images.delcampe.com/img_large/auction/000/128/963/860_002.jpg?v=1
https://shepquest.files.wordpress.com/2013/11/acs-scenes-1.jpg
http://www.playbill.com/news/article/playbill-archives-les-miserables-1987-119301