I rummaged around a little without really finding much about your ancestors that would reveal what was going on in their lives during the war, but I have a few ideas.
The 1860 census of W. C. Byrd's family at or near the village of Brunot shows that the 44-year-old head of household and his 30-year-old wife moved out of Kentucky nine or ten years before the 1860 census. I say this since 11-year-old son William C. Byrd was born in KY but 9-year-old George W. Byrd was born in MO. A telling factor to me is the large size of this family with six children in 1860 between the ages of 11 and a toddler 11/12 of a year old. I say this because the hamlet of Brunot in the southern tip of Iron County was several miles away from the large Union army base at Pilot Knob. In other words, William Byrd was living in the middle of a region rife with guerrilla war and the comings of goings of military men with a large, vulnerable family. I wonder if he felt the pressure to sell this place and move his large brood to a safer place at least until the war ended.
I also checked the MO Sec'y of State's website to see if the military service records there may have shown military service for William C. Byrd. They do not, but there is a record in August 1864 for a 19-year-old William Byrd who joined that year Captain Buxton's Company B of the 47th Missouri Infantry Regiment (Union) at the Union base at Pilot Knob. This William Byrd impressed his superiors enough to make corporal before he was mustered out 29 March 1865 at Benton Barracks in St. Louis. Your William Byrd's son William would have been about 14 or 15 in 1864. I wonder if he lied about his age and served a few months in the Union army, or was there another William Byrd in the region? You could complete a National Archives and Records Administration Form at your local library and send off to Washington, D. C. to get a more expansive copy of this corporal's military service record to be sure.
Just some thoughts.