Look back at the Dred Scott decision. If blacks were not and could never be citizens, then free or slave they did not deserve representation in Congress. So the South shouldn't have even had the 3/5ths total.
The population of the north had risen to the point that south had lost it's power to control and even be equal with the northern states.
And what in the Constitution guaranteed the South the right to control the Northern states, much less be equal to them?
Before there had been an unwritten power sharing arrangement in the executive branch: southern president, northern vice president or vice versa.
And what prominent Southern Republican was available to run with Lincoln in 1860?
If all they really cared about was protecting slavery they could have stayed in the union and helped pass the Corwin Amendment, or some other deviation of it.
The Corwin Amendment had a fatal flaw that the South would not have stood for - it protected slavery only where it existed and did not protect the expansion of slavery into the territories. Nor would any amendment protecting the expansion of slavery ever pass out of the House. So the South did leave, and the did leave to protect their institution of slavery, and in the process adopted a constitution that protected slavery in ways never imagined in the real Constitution. The confederate constitution specifically guaranteed slave ownership, slavery in the territories, slave imports, and most likely guaranteed that an amendment ending slavery was impossible to pass.