I can at least address half of it. There are three lines of pretenders to the French throne: Bonapartists, Legitimists (the Bourbon line), and Orléanists.

But a great distinction between any possible Confederate pretenders and French pretenders is that each of the French houses had laws for succession. They all use the standard French royal succession law, pure male primogeniture. They just have different starting points (and the Napoleonic law originally disinherited the lines of two brothers of Napoleon I).

For the Confederacy, the member states have not passed laws to provide for selection of electors for the Confederate States of America, nor have such electors met in their respective state capitols to elect a president, nor has the Confederate Congress met to count the electoral college votes.

There's another big difference between the two cases: in France, royalism of pretenders was a potent political force at times in the 1800s. I'm pretty confident that no Confederate pretenders have had any significant importance.