I can't say I have ever seen Dawkins get involved in a debate with a Christian apologist. Do you have one you are thinking of? I enjoyed the debate between Hitchens and Dr. Craig, different styles of reasoning again, but Hitchens repeatedly refused to take up the Kalam CA, because it was an impossible question to answer. I would love to see footage of Dawkins refuting the CA. I know Dawkins cancelled a debate with Dr. Craig in what can only be described as a snit. Too bad.
Dawkins has debated Dr. John Lennox twice. I admire Dr. Lennox though I don't really follow his work.
Hitchens really didn't do a very good job of defending atheism in that debate, but I wasn't expecting him to at all. I'm not the biggest fan of Dr. Craig, either, but it's always nice to see someone defending the faith in the public square. He's certainly a talented debater, and a grand theologian for cosmological arguments and the like, though I disagree with him on specific doctrine, like his rejection of divine simplicity and such.
I don't think Dawkins will refute the cosmological argument anytime soon, his attempt to refute St. Thomas's version(s) in
The God Delusion was pathetic. I don't know why he wouldn't engage Craig, who is obviously one of the most famous apologists alive, if he were sure his position to be correct. Perhaps he's afraid to lose (I would be, too) but I definitely wouldn't go around writing books on the subject if I weren't willing to engage in discourse.