It's possible that the characters concerned, the writers, perhaps even the target audience, are not so acquainted with that particular melody as the present company. Or perhaps Cox himself, with his sharp wit, has worked out that the abridged version can only accentuate its withering effect. He may have noticed that Carla was about to leave and had to cut the tune short in order to finish within earshot.
Or maybe he has something against Mojo somehow, and by some epic thought process realised that across the fourth wall, a million viewers, Youtube, search engines, video games, forums and the peculiar impulses of a couple of eccentric internet message board enthusiasts he could cause a farewell thread to veer so terribly off-course and off-topic.
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