I don't believe that was ever an issue. And there does seem to be a major difference between being on the side of a foreign country in a war and taking a side in a domestic dispute that represented over nine million Americans (the CSA states' population at the time.)
Regardless, I probably would ignore it. Rights are for everyone; it's too slippery a slope to let the government ever get its foot in the door deciding what is and is not acceptable.
The whole point is that the CSA claimed to be a foreign government and their supporters in the USA agreed. So they were by definition supporting a country at war with their own. IOW, it wasn’t a “domestic dispute” at all.