One source of info about other lyrics (verses) to "Lyria,The Tattooed Lady" is Michael Feinstein. He had a TV special (PBS, I think) back in the early 1980s, in which he sang a version of this song that included the line
"When she stands, her lap gets lit'ler,
When she sits, she sits on Hitler."

Feinstein was an apprentice to Ira Gershwin for some time and that, apparently, was one source of his "inside information" about songs and songwriters from the jazz era It will probably surprise no avid Marx Brothers fans to know that, according to Michael Feinstein, the original lyrics that Yip Harburg wrote for this song for Groucho were much bawdier than the ones that eventually made it (past the Hays Code into the movie.

Per Wikipedia (today):
"As lyricist Harburg explains, the final stanza originally contained the lyrics "When she stands her lap gets lit'ler/When she sits, she sits on Hitler"; the lines were removed because the studio feared the song would sound too dated. On some subsequent radio appearances, Groucho sang the song in its entirety. [Michael] Feinstein's recording omits a number of other lines, but restores the "Hitler" couplet ... ."
Marx Brothers: At the Circus

One thing that has long puzzled me about this song is who the heck "Mendel Picasso" was.
Apparently that's still a mystery, even to Yip Harburg's son Ernie:
Mendel Picasso.
I guess it just sounded good enough and silly enough for this song.