Well, according to the finding aid:
Copyright has not been assigned to the Performing Arts Special Collections, UCLA. All requests for permission to publish or quote from manuscripts must be submitted in writing to the Arts Special Collections Librarian. Permission for publication is given on behalf of the Arts Special Collections as the owner of the physical items and is not intended to include or imply permission of the copyright holder, which must also be obtained.
Upon reflection, I'm actually not sure of the legalities involved. I'll have to email the library and ask.