Searching on General Society of the War of 1812 message board, I got more information than ever expected. I found confirmation of how my 4th g-grandfather died and the site of the cemetery among other things. Never stop looking.

Jeremiah, Hercules, Moses and Jeremiah (Jr) enlisted in the US Army for five years on 24 Dec 1812 in Roane County, TN. They served in the 24th Regiment of US Infantry and they were stationed at Fort Meigs and later Fort Stephenson (on the Sandusky River) in Ohio. Part of the regiment was transferred to Buffalo in the fall of 1813. It appears that Jeremiah was wounded during the Battle of Buffalo and died at 11 Mile Creek where the US Army hospital was located. He would have been buried in the army cemetery in what is now Williamsville, NY.

All three sons transferred to the 1st Regiment of US Rifles and they were stationed at Sackets Harbor, NY, for the remaining days of the war. After the war they were transferred to the Michigan Territory. They were discharged on 24 Dec 1817.

p.s. The two youngest, who enlisted at ages 12 and 14, both died in 1833. Probably from an epidemic. Hercules died on his farm outside Dover, TN in 1868. They had 11 siblings.