Heck---If you like the Spiller and Burr go for it!
Griswold and Gunnison, Leech and Rigdon are other Southern produced
guns that are available as (somewhat) true to form replicas.
An artillery private in "The war of Northern Aggression" could have carried
almost anything that went bang as a sidearm, or carried nothing at all.
It would be a gun purchased with their own money or captured on the battlefield, so anything made in that time period or before would be "correct".
ETA: Here's an interesting link http://civilwarhandgun.com/obscure.htm
A few on there I've never even heard of----