To answer my own question: I find that Battery C, 1st Battalion Tennessee Light Artillery, had Capt. Jeremiah C. Martin, and 1st Lt. James H. Kile. Seems to be the likely identity.
James H. Kile enlisted in Battery D, 11/1/1863, age 33, 1st Sergeant 12/31/1863, 2nd Lt. 9/20/1864, 1st Lt. 3/6/1865. Also a POW but no info.
Jeremiah C. Martin Captain Battery C. Captured at Rogersville, (east) Tennessee 11/6/1863. confined at Macon, Georgia and Columbia, SC.
Battery D Light Artillery
TENNESSEE
(3-YEARS)
Battery D was first ordered to report to Brig.-Gen. R. S.
Granger "for duty with the artillery of Nashville," on March
23, 1864.
It was afterward transferred to the District of East Tennessee
and on April 2 1865, was assigned to the 4th division,
Department of the Cumberland. It was relieved of duty on June
26, 1865, and soon thereafter was mustered out at Nashville.
Source: The Union Army, vol. 4, p. 390
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Battery C Light Artillery
TENNESSEE
(3-YEARS)
Battery C, in the organization of troops in the Department of
the Army of the Tennessee, commanded by Maj.-Gen. William T.
Sherman, Jan. 31, 1864, was stationed at Fort Pickering, in
the brigade commanded by Col. Ignatz G. Kappner.
It was later transferred to the Department of the Cumberland,
and on May 31 was in the District of Nashville under command
of Maj. Josiah W. Church. It remained at Nashville until
March 22, 1865, when it was relieved from duty at that place
and assigned to Johnsonville, Tenn., where it remained until
the close of hostilities and was mustered out.
Source: The Union Army, vol. 4, p. 390