As Pushmataha had no relatives that we know of alive by 1900, the only DNA would be from his sister, Nahotima. my ancestor. I object very much to the ascertian you have DNA evidence of this, as there is no YDNA that can be verified from Pushmataha, and someone has linked my family to this.
Simon Favre did have family among the choctaws, but his wife was a daughter of Franchimastubbee (I have a testimony to that effect).
it looks like someone has made an assumption on the DNA without proving a link.
I hear your objection, and the answer to it is above my pay grade. :)
I've been in contact with the historian Rachael Roby on the matter in the past, and I believe she has done some triangulation of DNA results that points to the Moore connection, but as you pointed out, no direct chromosome evidence, only signposts to the Moore family and likely Gov. Moore, to my limited understanding. The connection is intriguing in part since Gov. Moore, as I recall, did a large trade in enslaved American Indians, which was illegal. The Moore family is therefore penciled in on Geni as in my tree.
Here's are the links to the info I have on the matter:
Pushmataha, Principal Chief of the Choctaw Nation
Hope this helps!