Playing devil's advocate for a bit:
I think the bigger issue than 'alternate history' is actually the idea of taking it in isolation from the rest of the world. The Man In The High Castle doesn't neglect the clash of Germans, Japanese, and US as part of its world history. But if you say "third civil war" and times has actually moved forward since the first one, then you're talking about WW1 / WW2 style combat, and not the mostly isolated war arenas of the civil war era (which was close to ending at that time anyway, but a WW1 era civil war would have been a very different war).
Typically speaking, that kind of no longer isolated uproar means the state is weak and ready for invasion ( 'benevolent liberators' of course) by other states, so in that scenario the US would likely be invaded by England under the guise of Canada and potentially some re-taking of the Southern states by Mexico as they were taken from Mexico earlier. If the opportunity presented itself, so close to the US-Mexico "war", there would be no reason for Mexico not to try to retake those areas, regardless of the potentially even more disastrous outcome (a world without Mexico, that is. An alternative 'slavery' scenario is not limited to African Americans after all, even if the real US history mostly has been, afaik).
So there might be a world of difference between the pitched premise and what that premise would lead to once they actually have to work out the design of that world without it immediately becoming unrealistic (that is: a state of continuous civil war somehow putting the US in the same position it is today, which would be ludicrous. It would be a lonely nation on the globe, considering abolition was an ongoing trend in Europe even before the US civil war. The idea of the world simply accepting an ongoing confederacy based on slave labor is a hard thing to sell as realistic).
Also, a state in turmoil could obviously not mount a similar defense as one that isn't, so if the time period were to reach the world war era, it would be the Soviets 'saving' Western Europe and then you have C&C: Red Alert potential. Just saying.
