You might try to send to the National Archives and find out if there was (a) a military record, and (b) a pension record. IIRC, a Springfield muzzleloading rifle-musket M1861 or M1863 went for about $13.00, which was at least a month's pay for a private. Unfortunately, I don't have any idea the cost for a Burnside carbine. Soldiers mustering out of the Union armies were permitted to purchase their arms on mustering out. These carbines are also referred to as '4th Model, 2nd Type'. The modification consists of the small screw in the right side of the receiver, which projects into a purpose-made groove in the breechblock, for the purpose of smoothing the motion to open and close the action. This is a collector's convention, referring to the last modification to the Burnside series carbines. While I haven't had time to look up my references on the subject, just a short note about the term '5th Model Burnside'.