While I haven't had time to look up my references on the subject, just a short note about the term "5th Model Burnside". This is a collector's convention, referring to the last modification to the Burnside series carbines. 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. These carbines are also referred to as "4th Model, 2nd Type".
Soldiers mustering out of the Union armies were permitted to purchase their arms on mustering out. Unfortunately, I don't have any idea the cost for a Burnside carbine. 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.
You might try to send to the National Archives and find out if there was (a) a military record, and (b) a pension record. The latter are often more valuable and interesting in finding out about a particular veteran's service and post-war life, depending on how long the veteran survived after the war. In some cases these pension applications were filed by the veteran's widow. Where the vet himself filed the application, sometimes in the 1920's, they often listed medical conditions that afflicted them. A lot of times, they were inclined to present proof of their disabilities to include hernias, and even ingrown toenails, in addition to more, ah, pertinent problems. Most of the time, they got the pension. The important thing is that some of the details of their service offer quite personal insights to the Civil (and Indian) wars.