I think it could work, but agree it would need to be camp and over the top. I think something similar to Xanadu would be ok -where the cast and audience are in on the joke. Maybe just off broadway though - not sure if it would have enough mainstream appeal to last long on Broadway.
I would have thought that if a musical was based on the movie and not the book then it would be the film studio who would need to give the rights and not Christina??