Amazon shows the song recorded by Eddie Crow and by "Guitar Shorty" but both are blues performers and likely used the Dr John version. Also done by Les Brown, but that was shown only in a listing with no other info.

Most places seem to think it's exclusively a Dr John song, but Country Titles shows Buck Owens as the Composer and Dr John as performer. No other info at the site though, so that may not be a reliable bit.

Also seen with the shorter title just "The Dog" but that doesn't show up in any Owens associations either.

Owens appeared on "Hee Haw" and also on "The Jimmy Dean Show" somewhat earlier, so it's possible the song - in his version - might show up in old tapes of those shows if you can't find anything else; but I don't see a way to search them very efficiently.

John