'The Barnum Family, 1517-1904' calls him Zadock Barnum. He served in the Civil War, as a Private in the 1st Wisconsin Light Artillery.
Duane E. Howard says he was probably killed in the Civil War. His brother Noah was in the 16th Wisconsin and did die as a result of wounds received in battle.
He was buried in Chalmette National Cemetery.
Zadoc Barnum is buried in Chalmette National Cemetery, where his listing reads: Barnum, Zodac [sic]; sec. 151, grave 12405; Pvt.; 1st WI Lt. Arty.; d. Apr 3, 1902; Civil War.
Chalmette National Cemetery is a 17.5-acre strip of land that sits adjacent to the site of the Battle of New Orleans, along the Mississippi River in Chalmette, Louisiana. The cemetery, which is closed to future burials, has more than 15,300 interments from every major U.S. conflict between the American Revolution and Vietnam. While most of the individuals interred are veterans, civilians such as spouses, children, and employees of the War Department are also included.
The cemetery was established in May of 1864 as a final resting place for Civil War dead, both Confederate and Union soldiers alike. Approximately 132 Confederate prisoners of war were buried at Chalmette until the Ladies' Benevolent Association of New Orleans requested that they be moved out of Chalmette, which is comprised entirely of Union soldiers, to the Greenwood Cemetery in New Orleans. In the years following the war, the cemetery took in an additional 7,000 interments of Union soldiers moved from abandoned cemeteries located all over southern Louisiana, Ship Island, and Mississippi. Civil War burials at Chalmette number well over 12,000, of which almost 7,000 are unknown.