Your brass flatbutton's backmark says "Fine Gold Color." Since unlike most flatbuttons it doesn't have the British spelling (colour), it is American-made... and therefore, was manufactured sometime between about 1815 and the early-1830s.
Actually, after closely re-examining the first photo, I now think it does have the British spelling, "colour." So, it is from the after anti-British boycott after the War Of 1812 diminished, likely mid-1820s into the early 1830s.