Ebenezer died as a result of wounds suffered during the Battle of Fort Erie (Ontario, Canada), which took place on July 13, 1813. In American Genealogy Magazine, Vol 13 #4, Dec. 1998, there appeared an article titled 'War of 1812 Widows & Orphans abstracted by Craig R Scott, C.G.R.S. Part VIII'. Within that article, on page 16, there appeared an extract of New York records, which includes the following: "Box 13 - NY; 1851:03; 5 Sept 1818; Tioga Co, NY; Ebenezar B Barnum; Polly Barnum, Widow." That entry would seem to indicate that Polly (Everest) Barnum applied for a pension as the surviving widow of a soldier killed in action.
According to family lore, Ebenezer's wife Polly was so distraught at his death that she renamed their son (original given name not known) to  Ebenezer Benedict Barnum, Jr.