I'm not sure what "supremacy issue" you think was there, (I'm not aware of one) but the split, totally engineered and premeditated at the Charleston convention by the fire-eaters, broke the dems along anti-secession Douglas Democrats and pro-secession Brackenridge democrats. Other than secession, their platforms were nearly identical and aside from slavery expansion, the Republican party platform in 1860 was not radically different than either of those two. All the party platforms, for instance, supported Federal monies for a trans-continental railroad. The only disagreed on what state the Eastern terminus would be in. The only significant issue in the 1860 election was the question of the expansion of slavery.
The fire-eaters engineered the split to assure a Republican victory that would virtually guarantee the secession of the deep-south. Not even they would have gone along with secession if a Republican had not won. The "Black Republican" propaganda broadcast by the fire-eaters in the years before 1860 sufficiently terrified enough of the non-slaveholding population of the south to go along with the secessionist slaveocracy. Read the "Black Republican" propaganda of the day and you will see that the Jessi Jackson/Paul Bugala/DU lies and hate mongering we see today are among the oldest traditions of the Democrat Party. They have always preyed on the uninformed and ignorant. It’s what Democrats do best.