Your brass 1-piece flatbutton find is a US Army 2nd Infantry Regiment Officer's button, made in Philadelphia by George Armitage, sometime between 1808 and 1825.
It is shown in the button-book by Alphaeus H. Albert as button GI-53-R2-G.
The McGuinn-&-Bazelon book on backmark time-dating says George Armitage was a "plate worker" who immigrated to the US from Sheffield England. His US Officer buttons were made of what is called "Sheffield Plate" -- which one of the earliest versions of Silverplating. Hence the term "plate worker" for Mr. Armitage's profession.