A Little Priest and Bosom Buddies immediately came to mind.
I would also add The Grass is Always Greener, Class, second the previous mention from Gentleman’s Guide (in fact half the score) is funny. From The Producers, I would propose The King of Old Broadway, Betrayed (to me, the comedic Rose’s Song), and Keep it Gay.
Others would include Little Girls, Gooch’s Song, Broadway Baby. Comedy Tonight sets up Forum, but the song itself isn’t very funny. Everybody Oughta Have a Maid is a funnier song.