Here's a recording that calls it "Alvin the Alligator":
https://www.youtube.com/watch?v=jbwUaCUnyjQ

and another:
https://www.youtube.com/watch?v=38NNawUZ4Ho

Another recording: http://thebirdsings.com/songs/alfred.html (Alfred, as a round)

I think I found the author's name. One source I found said it was by Peter Tork. This Monkees Book says the song was titled "Alvin," and was written by Peter Tork's brother Nicholas Thorkelson. Recording can be found on later editions of the 1968 album, "The Birds, the Bees, and the Monkees."

-Joe-