All of you are completely misunderstanding what kind of man John C. Breckinridge was. He was not a Fire-Eater (and did not much associate with them socially, either), opposed secession, and was himself quite ambivalent on slavery. By 1860, he had not been a slave owner for many years. Many members of his family were strong abolitionists and Breckinridge himself had been a subscriber to Frederick Douglass's newspaper. In many previous political campaigns in Kentucky, Breckinridge had been attacked as being anti-slavery, and even in the 1860 Presidential Campaign the John Bell supporters had attacked him for not being a slave-owner.
The only defense Breckinridge ever made of slavery was that the Constitution protected it; he never argued that it was moral or economic. The only defense he ever made of secession was that it was a constitutional right; he did not think it was a good idea at all. His political heroes were Daniel Webster and Henry Clay, not John C. Calhoun. He was a bridge-builder and a compromiser, not any kind of pro-slavery radical.
Breckinridge would not have used his years in the White House to push a pro-slavery agenda. He would have striven to hold the Union together by any means he could. Having come into office in such a backhand manner (which would have utterly horrified him, I'm sure), he would have certainly filled his Cabinet with moderates from all regions, and probably even Republicans if he could persuade them to come in. He would have fiercely resisted any effort to use the Dred Scott decision to allow slavery into the Northern states. In the scenario you are outlining, Breckinridge would have bent all his energies to achieving some sort of compromise with the Republicans on the question of slavery in the territories.
Breckenridge wasn't a fire-eater, but he certainly checked off all the boxes for the standard Southern pro-slavery position. He did not believe Congress had a right to restrict slavery in the territories; he had endorsed the Kansas-Nebraska Act as well as the Lecompton Constitution, he had supported the Dred Scott decision, opposed the Homestead Act as well as believing in a right to secede. That last point is one that is sure to make the 64/65 transition worse than the 60/61 IOTL.
But I do understand your point about him though, he would probably turn out to be four more years of Buchanan. So would Lane be a better candidate for maximum turmoil? I don't know much about him beyond him openly sympathizing with the CSA, using his last day in the senate to berate Andrew Johnson for opposing secession.