This is really intriguing to me but I can't find anything about it. I really have my doubts the Carter Family did it -- they tended to be a lot more apolitical than that. Plus, like you said, I couldn't find the Carter Family singing it.

I did also find this in a PDF about Merle Travis:

As a kid, he absorbed the rich musical culture of his region. “There was music in the home, of course...then there was a fellow named Colie Addison who played the fiddle and he played the guitar and the old ‘tater bug’ mandolin, and that just sounded the purtiest that I ever heard, to me. And of course in home, whymy dad was a five string banjer picker. But he didn’t have a banjer and he talked about the old time banjer players... I remember he used to talk about a guy named Jim Winders who was a great banjo player. So finally my dad’s brother, named John, Uncle Johnny Travis, he got a five string banjo and Dad traded him out of it and brought it home and Dad, he’d play... pick it, you know, had two different ways, he called it knockin’ the banjer and then pickin’ the banjer. He’d sing songs... he’d sing ‘Jenny Weaver,’ and a song about

Jeff Davis swore when the cruel war begun,
I wouldn’t be the Union man and carry the Union gun,
But I’d rather be the Union man and carry the Union gun
than to be the rebel, the rebel had to run.

“That was the words to the song he sung. And of course he sung some little old verses to ‘Ida Red’ and a bunch of stuff.

Source (PDF will download on click)