It would be embarrassing for her. The audience is there to laugh. Carlin will score more on the laugh meter (for the typical Leno crowd), therefore, he will win. Coulter is probably well aware of this and chooses not to subject herself to it, any more than Carlin would dare go on a forum where he's scored on his grasp on reality.
Then Carlin or no Carlin, what is Coulter doing there? She going to do a stand-up routine? Or just looking for another easy appearance to shill her book?