Katy Perry is listed in the credits for The Warbler album, according to AllMusic.com, as composer. She appears to have composer rights to the song, despite being one of five listed composers (it took five people to write it?). So she'll get credit and all the perks that go with it no matter who performs it or how they perform it, apparently.
But you're deliberately being dismissive of this. Coulton didn't just sing the song again- he put work into it and revised it.
Cover versions of songs don't have copyright protection in the US, regardless of what the artist does with them. If it can be shown that an artist's cover of the song is sufficiently different from the creator's copyrighted version, it should likewise be able to be protected.
Fox has found a nifty loophole- they can now take arrangements that cannot be copyrighted, and only pay whomever holds the song's rights. Which is, in my opinion, just legal thievery.
"...everyone finally shut up, and the audience could enjoy the beginning of the Anatevka Pogram in peace."