And then Davis would have sacked him. Davis kept appointing non-entities to run the War Department so that he could be personally involved. IIRC he tended to try and run the government like the army, everyone taking his orders. He only allowed other people to run the war when he didn't have any choice IE 1865 with the War already lost.
That may not be true. Davis respected Breckinridge, and might have been inclined to listen to him, as he was to Judah Benjamin (another man he respected).
One possible butterfly to come out of all this...Breckinridge might not recommend that the Cleburne Memorial, when it comes into his hands in January 1864, be squashed. Breckinridge was one of the supporters of the black recruitment measure which was finally passed in 1865, and authored General Order 14 (which specified that black recruits had to be freed before being accepted into the army, that black recruits were not going to be formed into segregated regiments, and that black recruits had to receive equal pay, rations, clothing, and treatment). So it is not beyond the realm of possibility that he might even endorse the measure, and urge Davis to act upon it. Whether Davis would listen, of course, is unknown. But in OTL it was partly James Seddon's (the OTL Secretary of War in January 1864) horrified reaction to the Cleburne Memorial which influenced Davis to treat it as he did.