Hello Jimmy:
Tried to look up that exact button without success but here is some information that may help you to find out more about it.
That type of cross is called a cross formy or cross pattee.
Crosses with a point like that are called cross fitchy - any cross whose lower point has been converted into a long point. Therefore the cross on that button is called a cross formy fitchy or a cross pattee fitchy.
It may not be military at all-maybe made for a Masonic Group or something like that. The symbolism is similar to Knights Templar but most of the crosses I saw were not the exact formy type.
George Armitage made buttons in Philadelphia on Race Street between 1799 and 1826. There may be a catalogue somewhere listing all of the buttons made at that factory. Lots of experts on this forum so somebody should be able to get closer to identifying it.
There were plenty of military activities during that time period in the USA including the War of 1812 but I couldn't find anything resembling that button in the military.
Good luck with finding out what it is. It is a very interesting button because of the size and symbolism. I will watch this post to see if anything comes up.
Foiled Again