If we go with the OP, the north will essentially have 80-90 % of its GDP OTL and the south will be, well, poor and defenseless. They'd be overrun in a year.
They could still build some nice forts along the rivers in the Upper South/Texas and maintain a sizable riverine fleet on the Tennessee and Mississippi and any sizable river in Virginia. The Mississippi and Tennessee rivers and the Appalachians will channel attacks down particular angles and prevent armies from easily assisting each other until river crossings can be secured. Foreign support could help modernize their army to a degree although it's going to be pretty dated in many places. If the South intertangles itself in foreign alliances, it'll probably be prepared to mount a solid resistance. I'd expect Canada to be an easier nut to crack in this scenario, simply because even with the Canadian winter, taking Montreal and Toronto is going to be easier than taking major CSA centers like Birmingham or Atlanta.