Your analogy would hold if the CSA had claimed to be the legitimate government of the USA. However, the CSA never did any such thing and it acknowledged, at all times, that the government of the USA at the time was the legitimate government of the non-CSA States.
Going back to your President example, the Pope claiming to be the head of Orthodoxy would be like GWB claiming to be the Prime Minister of Australia.
You've got it backwards, of course. The Pope holds Peter's authority and office. You split off from the Catholic Church, not vice versa. Thus, in the analogy, Lincoln is like the Pope and Lee is like the orthodox patriarchs or Luther or pick your favorite schismatic.
None of the ancient sees established by the apostles survive unbroken to this day except for Rome, the ancient see of Peter. All the patriarchates are beneath Rome in terms of authority as any perusal of the first three hundred years of Church history easily shows.
You can claim what you like about Constantinople, but there is absolutely no theological reason to believe that Constantinople has any claim for authority that competes with Rome's. The closest you can get is to say that a Roman Emperor had his capitol city there for a time - that's not theology, that's just interesting cultural data.