I've no doubt of that, Steve. I've usually heard it sung "wooden shoe," not "wouldn't you." The sloppy or ambiguous pronunciation is the whole point of the song.

Sheet music or liner notes are often discrepant from what's sung on the (common) recordings. Sometimes whole words are omitted or added.

The fact that your sheet music does not include additional verses that are also on the 1940s recordings is also not unusual. Sheet music sometimes omits a song's introduction and/or additional verses.   Sheet music also may or may not indicate the same instrumental arrangement that's used on the recordings.   

However, checking out some old recordings of the song, I'm hearing "wouldn't you" more often than "wooden shoe." Perhaps as the song made its way into our arsenal of "children's songs" -- which it was not, originally -- the "wooden shoe" pronunciation became more common (because it fits the theme of the mumbled-word verse). In any rate, I prefer it that way.

Mairzy Doats: The Merry Macs, 1944They sing "wouldn't you."

Spike Jones: Mairzy Doats Only one verse, repeated. Sometimes it sound like "wouldn't you," sometimes "wooden shoe," and sometimes like "wooden Jew."

Pamela Holt: Mairzy Doats She also sings "wouldn't you."

Some of the various recordings I've found not only pronounce "wouldn't you" as such but also articulate lyrics like "kid'll eat ivy" and "little lambs eat ivy" too clearly, to the point where they don't sound as "queer and funny to your ear" as they're supposed to.

I'm trying to find Kay Kyser's recording. Maybe that's the one that had the additional verses.