It's not an exploit, really, or at least nothing as serious as a security hole.
He just logs out really fast and, because of the way Livestream is set up, you cannot ban offline users. The "You cannot ban a chat moderator" is just the generic "Banning failed" message- he's not really a mod, spoofing, or anything of the sort. If you click fast enough, you can ban him. Our channel has 26 instances of this same guy in our ban list as of this posting.
Also I notice a lot of you are not clearing his messages. As long as you don't refresh the page, you can still click the "ban" by his name and clear his messages- even if he's already offline. Still really annoying, but it lessens the effectiveness of the spam and is better than resorting to log-in requirements.