The lyrics sound interesting and it would be nice to find a tune. Furthermore, I don't think we've nailed down the origin of those lyrics.

The US catalog of copyright entries lists 2 songs:

I'VE LOST MY DOG, words, Irene Franklin; music, Burton Green. New York: Leo Feist, Inc., 1926. [The British Library has the sheet music.]

I'VE LOST MY DOG, words and music by Jerry Buckley, Detroit: Jerry Buckley, 1926.

A Google search found a book:

Funny Songs for Children by Burton Green and Irene Franklin (Melbourne: Allan & Co., 1926.) which contains I'VE LOST MY DOG.

The Online 78-rpm Discographical Project lists 3 recordings:

Howard Israel, Al Hall, and their Peerless Serenaders, I'VE LOST MY DOG, Okeh 40729, 1926.

Ray Miller & His Orch, I'VE LOST MY DOG, Brunswick 3287, 1926. (Composer: Jerry Buckley) [The University of Missouri at Kansas City has the recording.]

Davey Lee, I'VE LOST MY DOG, Brunswick 4986, 1930. [Note: Davey Lee would have been about 6 years old when he recorded this.]

I can't find any confirmation of Foolestroupe's assertion: " 'Lost My Dog' was a jingle written for Pepsi by Hummingbird Productions" – and even if that's true, I doubt that it's the song we're looking for. The lyrics we have seem awfully long for a commercial jingle, and I don't see any tie-in with Pepsi.