I don't by it. All planet-moon systems orbit the barycenter and not each other.
That makes sense when you look at the Pluto system Barycenter which shows all the objects orbiting the barycenter.
But then, think about the Earth-Moon Barycenter. The Moon is really not orbiting the Earth. Both the Earth and Moon are orbiting the barycenter.
It makes more sense to me to have Charon stay Pluto’s moon.