Can anyone tell me if it was ever recorded, on what album - and perhaps even where I can purchase it? The band you're looking for is named "Corky and the Juice Pigs", the name of the song is Eskimoand it's on their first album, sometimes called The Debut Album. I can at least say that with authority. However, as a standalone song, the Mad TV version of 'Eskimo' is better, as it features more impressions. Dr Demento's 30th Anniversary Collection. Oh, and Tenacious D has absolutely never done a cover of it as an album version. I've not actually heard a Tenacious D version, and every version I have actually seen online has been mislabelled as such when it doesn't even sound like Jack Black's voice.
Oddly enough, a different webpage has the same track list for that album, minus the gay eskimo song:
One guy tags an MP3 wrong and shares it, and suddenly people can't tell what band sings what The song is on a CD I have. I believe this it the version you're looking for https:
However, as a standalone song, the Mad TV version of 'Eskimo' is better, as it features more impressions.