Actually, it is not a "Federal" button... according to the button-book by Alpheus H. Albert, it is a generic Militia button, used by various militia units in several states. Albert shows your button as GI-62, in the Militia section.
About your button's backmark:
The firm of Lewis & Tomes in Birmingham England was a button-dealer from 1816 through 1830. The War Of 1812 caused much resentment in the US toward Britain and British products, lasting for more than just a couple of years after the war's end (in 1815)... so your British-made button for American militia units was most probably not made and imported into the US until the mid-1820s.