What I find interesting is that Mallett's has staged this as if Frazz created the joke for himself, and unintentionally started telegraphing the punchline in panel one. The Hallelujah chorus that is perhaps the most famous portion of Handel's "Messiah."
Interstingly, "Messiah" was not written as a Christmas tune. It was indeed written as a tribute to Jesus Christ, but it is meant to reflect his entire life, not just his birth. In fact, it's Part I of the song that references his birth. Part III, in which the Hallelujah chorus occurs, covers the resurrection of the dead and Christ's glorification in heaven. So that portion might be more appropriate as an Easter song, rather than a Christmas one.
Personally, I'm not overly partial to religious jokes, and I think this one isn't delivered especially well, compared to Mallett's usual punchlines. But I do find it a fascinating bit of character exploration that you don't normally find in newspaper strips.