While the version on the left is (from ItsyBitsyKids) I originally encountered this song from the Amidon's amazing "I'm Growing Up!" collection. I have heard it at a few Orff workshops (but I can't recall if it was presented or if participants were just sharing amazing, no-fail resources.) I emailed Jody Kruskal, the source of the song in the Amidon collection, to find out where he learned it, but I haven't heard back yet.
It shows up in the Wikiversity nursery rhyme collection, without a source or date. All Nursery Rhymes.com states that it's "from the 1950s" but does not have a source, and all of my usual nursery rhyme sources haven't shown any sign of it, other than in a children's book by Martha Mumford (2021) and in mentions in a motor skills/sensory skills development book published in 2020, all from the 2000s and later.
I tried Mama Lisa's collection and found a German folk song titled Häschen in der Grube (Rabbit in Your Burrow). Mama Lisa gives credit to Friedrich Fröbel (the inventor of the term "kindergarten") on one page, on another page she credits Friedrich with the lyrics and Karl Enslin with the melody, and on another, it claims the melody comes from Fuchs, du hast die Gans gestohlen. I posted both melodies below - they are very similar.
I took the melody to the Folk Tune Finder, and came up with ES LASSEN SICH DIE TODTEN FUERSTEN BALSAMIEREN, which translated to "THE DEAD PRINCES CAN BE EMBALLIZED." I found it on Musescore as well, but could find no other information. The melody has similar features, but the children's song cuts out a lot of the repeated phrases. My lack of German language comprehension means I have hit a dead end on this one!