Corrupt Overseers want him out because he's in their way of gaining more power and potentially even getting into the big chair themselves. I sincerely doubt Campbell didn't make any enemies in his selfish sacrilegious blasphemous climb to the top of a religious order.
Pure Overseers know full well he's a corrupt snake and are glad to see such a man removed from power. Maybe if he were a good, righteous man who followed all the strictures perfectly someone might stick up for him, but given he breaks all of the strictures EVERY SINGLE DAY I doubt anyone would even bother. I'm not sure if they believe in any kind of God or Karma or something but if they do they probably ascribe it to that.
Either way, it's not like they can remove the brand; the damage is done.