There is a difference between local necrosis and systemic sepsis. That black area will eventually get bad enough that the infection gets into the blood stream and sets off a inflammatory chain reaction that will kill the animal. That is when you see lethargy, and other end-stage signs of systemic infection. I anticipate the only way to save the animal is to amputate the tail, similar to gangrene in a human. It will be hard for antibiotics to do their job with so much internal damage. No way to deliver since the blood supply is compromised. If we can see see a large black spot through the scales, the damage underneath is likely massive. Once this gets to the GI tract, it will likely progress very rapidly.
This needs to be a lesson learned to everyone the seriousness of stuck shed on the tail. People freak out about a retained eye-cap unnecessarily, but the necrosis caused by stuck tail shed can actually kill if left untreated.
Please keep us posted as to what happens, but I will be honest and say it will be a VERY uphill battle.