Here are the songs I can find with "codfish" in the title. (I have no information about them other than what you can find by clicking the link.)

AT THE CODFISH BALL, recorded by
Joyce Walsh (with E Kirkeby) in 1936
Mae Questal in 1936
Tommy Dorsey & His Clambake 7, in 1936.
Johnny Johnson & His Orch, 1936

YOU WANT ANY CODFISH, Billy West, 1922

I WANT CODFISH, Billy West, around 1923

WANT ANY CODFISH, Billy Jones, about 1923

HEY YOU WANT ANY CODFISH, Furman & Nash, 1923

HOME DEARIE HOME/THE CODFISH SONG, Mordy Bauman, 1946

THE CODFISH, a traditional song sung by Nora Cleary on the various-artists album "Voice of the People, Vol. 7: First I'm Going to Sing You a Ditty" (It sounds like a version of THE CRAYFISH

THE CODFISHER, written and sung by Casey Neill on "Memory Against Forgetting", 2005. This from a sound sample:
"I thought this would be the work I'd do until the day I'd die
But out beyond the Grand Banks where my small craft cannot go,
Out upon the ocean...."

DON'T KISS A CODFISH, performed by Tom Knight on "Don't Kiss a Codfish – When I Grow Up"

CODFISH SHANTY which sounds like a variant of THE HARRIN'S HEID [Herring's Head]