A friend of mine who is a keen collector of rounds is also still very interested.

There is a track ("previously unpublished") titled "Alvin" on a Monkees sampler CD, sung by a single singer a cappella - presumably meant for a "looper", either tape or software, which would render it as a round of 16 parts. (Did it myself, with software: great fun!) There is no indication that the Monkees ever claimed authorship, rather to the contrary. See Tork's song list.

"Alvin the Alligator" is the title of a children's book about tooth hygiene. Peter Tork or someone else may have transferred the name erroneously.