My first step would be to get a Colt Letter
Unless you have provenance or can access any existing Quartermaster ledgers, you'll never know 100%
The man whose name is on the frame may have bought the gun himself during the war and carried it
The name was most likely applied if it was a gift from family before he left for the war, or it was put on after he got out. Or he scratched it in during his time in
Stuff like this is always a crap shoot because information is so thin . You'll be playing maybe, what if , and probably without hard provenance
Colt can basically tell you where the gun was shipped, and in what configuration
It may be an Army depot, it may be a hardware store, or anywhere else. It will be a good start