No need to even send a Corps, honestly; the French Navy alone could probably deal a death blow by breaking the blockade of the South and forcing the evacuation/surrender of coastal Union positions. That would free up a lot of Confederate manpower, allow for all of the above and general trade (increasing the diplomatic chance of general recognition), and, finally, would prevent things like the Overland Campaign given the French and C.S. Navies would prevent the logistics needed for it. Whether or not they could've instituted a blockade of their own upon the Union is possible, although I am not sure.