Why would Lincoln be exiled? Is this thing claiming that the Confederacy took over the whole USA? That's not how it would've worked. They wanted to be a separate country from the urban North, not to conquer it. It would've been more like a North and South Korea or Vietnam sort of deal. (Hmm, interesting that such schisms tend to be north/south rather than east/west. Maybe because north and south parts of countries are more likely to be in different climate zones and thus have differences in economics, livelihood, and culture, like how the Southern US states were more agrarian than the North.)
Anyway, the US was one of the last countries to outlaw slavery, so I think if the Confederacy had become an independent slave-economy nation, it would've been under a lot of political and economic pressure from other countries to end slavery -- trade embargoes, sanctions, that sort of thing. I'm not convinced they would've been able to maintain slavery to the present day, because the US has never existed in a global vacuum. And because the slaves they still had would continue resisting or escaping to the North.