That's a good point, as well - unless Bran being dead, north or south of the wall, is even more important?
The way I understood it, the magic protecting the cave was breached because Bran made himself present to the night king while he was in it. Brandon isn't a walker nor is he dead so I don't think there is an issue with him crossing the wall, I think the Wall is coming down in the next few episodes.
I also question whether there could be a wight Dragon. The army of the dead is animated by ice while dragons are the personification of fire. It would be analogous to a red priest reviving a white walker with fire.