I had read someone talking about the button being found at Perryville, Ky and was not at all sure of the description as it being from Arkansas Military Institue or not. You have done plenty of study so I know that you know where in of you speak.
I saw a few years ago an Arkansas Militia Button, with the old Arkansas States (Double Eagle) seal, on display in a case of state buttons at the Gettysburg NMP Museum. They had removed the case last year when I was there for their remodeling. I sure wish someone would make some repro's of these buttons.