It was Lee's intent after the War to retire from public life. However, even in defeat, he was too much a celebrity. Washington College offered him the Presidency of their school, and he accepted. As President there, he is acknowledged as doing quite a bit of good for the school.
Now, most of what good he did for Washington College were things that would not translate well into politics. But one thing would hold true: Lee's name was the end-all in almost any scenario. He was so well respected that basically anything he wanted was going to get done.
So, really the question is how willing would he be to take up the mantle of President of the Confederation? For the most part, he'd likely do a good job. He'd be immensely popular. And, as President of the CSA, he'd not be in a position to take on huge responsibilities. I think he could be convinced to become President.
Unfortunately, there's not a lot of evidence from RL that would indicate how he would feel about the job. He was involved in politics after the War, but mostly as a spokesperson against Radical Reconstruction.