Three things with this scenario:
1. Although it was assumed that the VP took over if the President died, it wasn’t codified into the Constitution, and it hadn’t happened before a President took office. After the Civil War, this will be the first rule patch in the Constitution whether the nation stays together or not.
2. The only way the South wins the war is with outside assistance. Lincoln made sure the British stayed out of the war - if Hamlin is less smooth and the CSA calls upon the British, they have a fighting chance.
3. In addition to Kentucky and Missouri as slave states that didn’t join the CSA, the state of Maryland was on the verge of voting to secede. This means Washington, DC is surrounded by Confederate territory and will almost certainly have to be abandoned. After the war, the USA will need a new permanent capital. Philadelphia is possible but it’s built up and too close to CSA territory. One possibility is Chicago, especially after the Great Chicago Fire.