Well, ain't Google wonderful! Having written the previous post I thought why not just google it and see what appears. So dammit, now I have to unsay all that about it going back to the 1800's cause it turns out it was a song published as sheet music in 1905 by Vincent Bryan and Gus Edwards with the title "Tammany, a pale face pow-wow".
Still fairly esoteric though.

Except for a couple of variations, my version from memory was pretty close in most of the particulars, but of course as a little kid I never learned the additional verses, or perhaps my father just never sang them.

All of this makes me feel both great at being able to recover cultural knowledge experienced as a child, and extremely old, because who else alive now could possibly sing such an old ditty from personal memory? Yecch!!