Phil Harris did indeed record ELMER AND THE BEAR, as RCA Victor 78 rpm disk #20-3393, probably in 1949. On the flip side was THE MOUNTAINEER AND THE JABBERWOCK. This information is from The Online 78 rpm Discographical Project.

Phil Harris also recorded THE PREACHER AND THE BEAR, RCA Victor 20-2143. Lots of other people did too; I don't know whether Harris was first.