I think your best bet would be to see first if he applied for a pension, and under which name. You might find his name in a pension index at ancestry.com. You will need to write down the application and certificate numbers. I think I saw something about a Davison - not Dawson- Bowles with the KY 4th Mounted Infantry Company B.
The actual files would probably be at the National Archives if he did apply, and these files might shed light not only on his service but his reason for using the alias. He'd have to explain that to obtain the pension I'd think.
There should also be a seperate military file on him at the Archives, even if he never applied for a pension, which might give a monthly report of where he was. If he was away from the regiment as a POW it might have a notation saying, "April 1864, absent, taken prisoner at Chattahootchie River" or something similar.