The CSA, at the start of the game, is far from dominated by socialists. Its government, sure. But we are talking about the first weeks of a civil war. Thousands of those angry poor people you mention most likely voted for Long and his populist platform, thousands of others stuck to their guns and remained loyal to the Federal government. There is the National Guard which is not necessarily aligned with the state they reside in. Long's Minutemen are a nation-wide organization and are not limited to the Southeast - so their cells in all of CSA's cities would bust out and get into street fighting on the very first day. Considering that violence and riots were commonplace in the region, you would see Federal army units stationed there at the start of the civil war, too.
Oh, and by the way, which region was the KKK strongest in during the 1920s and 1930s? It's not the South. It's the very same Midwest which the CSA is trying to carve into a Syndicalist state.
What I'm trying to get at is that the first month, two months, three months of the war would be messy as hell. Russian Civil War levels of messy, where nobody has a clue who is in charge and which faction is in control of what. And the CSA does not have an army - it only has worker militias, which, if we take the Russian Civil War as an example again, are likely just mobs of workers armed with whatever they have and commanded by rebellious junior officers. No heavy equipment, no aircraft, no supply system or logistics.
Canada's professional army, which is canonically prepared for a Second American Civil War in advance and has drafts for action if it occurs, would eat the CSA for breakfast.