No, the Beatles did not perform “Happy Xmas (War Is Over).” But this video by a French Beatles tribute band, the Rabeats, gives one an idea of what the song would have been like as a Beatles song.
I like the idea of “Happy Xmas (War Is Over),” but two things about the song bug me.
First, it’s lazy; John took an old English folk song, “Stewball,” and wrote lyrics to mostly fit the melody.
And second, I don’t really like the performance. Some of the outtakes are better, and Yoko’s backing vocals really grate on me. I tend to like covers of “Happy Xmas” more than I like John and Yoko’s version. And this Beatle-esque version by the Rabeats, a French Beatles tribute band, is a particular favorite.
Unforunately, I could not remember for the life of me who recorded it. “The Rutbeats,” I thought, but that was wrong. I was in the ballpark, though, and with a little searching on YouTube I found it. The voices are a little broad, the accents aren’t quite right, but it’s done well enough that I forgive that.