In the scenario described, I think he would have done his best to persuade the Southern political leadership not to secede, working directly with Lincoln (who was a personal friend of his) to reassure the South that the incoming administration would not act against slavery. It would not surprise me if Lincoln offered Breckinridge a place in the Cabinet. Breckinridge would probably have not defended the federal forts and arsenals in the South, because he would have been very concerned to avoid bloodshed. Sadly, I don't think it would have done any good. The Fire-Eaters were determined to pull the South out of the Union and Breckinridge wouldn't have been able to stop them.
Although Breckinridge believed that secession was constitutional (if foolish), I don't think he would have recognized the Confederacy, seeing it as a decision to be made by the incoming administration.
Interesting. I wonder how the Civil War would be started - if at all - absent the Federal Forts and Arsenals.
The Diplomacy in the early part of Lincoln's term is going to be interesting to say the least.