Data from Indiana State University (and some other sources)

Title: PUDDIN' HEAD JONES
First Line: There was a most peculiar kid in our town, was always late for school....
Chorus: Puddin' head Jones was fat and funny, dumber than sticks and stones....
Music by: Handman, Lou. 1894-1956
Words by: Bryan, Al.
P/P/D: New York : Remick Music Corp., 1933.

allmusic lists 3 recordings on CD:

Joe Haymes, "Joe Haymes and His Orchestra, 1932-1936," IAJRC CD #1008, 2000.

Ozzie Nelson, "Very Best of Ozzie Nelson, Vol. 1: 1932-1934," Collectables CD #7477, 2002.

Don Redman, "1933-1936," Classics CD #553, 1990.

There are sound samples available, but I don't have time to transcribe them right now.

Anyway, the song is older than the dates you gave for the baseball player. Maybe the player took his nickname from the song. Or maybe there was another song.