“At the time of the war, the South claimed the North was fighting against the Constitution.”
At the time of the war, the South (i.e. the Confederacy) had repudiated the entire Constitution of the United States by withdrawing from the Repulic that the Constitution defined.
They were the ones fighting against the United States Constitution.
“At the time of the war, the South (i.e. the Confederacy) had repudiated the entire Constitution of the United States . . .”
Actually, the Confederate States adopted the United States Constitution almost in its entirety. The Confederates made a few improvements and clarifications based on experience but it was almost the same words that southerners had written and offered to the nation two generations earlier.
Even after secession, the United States remained the most powerful slave nation in North America, perhaps the world.