For whatever they are worth, here are my thoughts:
1. I think that Lee could have been persuaded to run for President if he thought it was an issue of duty to the new country and if he did not have to personally campaign for the office.
2. If the POD is a Southern victory tin 1862, 1863 or even early 1864, I think Lee's helath might be much better than in OTL since he would not have had to deal with the rigors of the 1864-1865 canpaign against Grant.
3. If elected President, I think that Lee's experiences in the war might push him towards more National type of policies for the new nation such as a military academy (VMI?) a naval academy (The Citadel in Charleston?), a true national army, an expansion of the Navy, support for railroads and modest industrialization.
4. I think the supporters of Lee might take the name "National Democrats", later shortened to just "Nationalists". The supporters of Davis and his friends such as Bragg might become the "Consitutional Democrats", later shortened to "Constitutionalists". They would consider Lee's poicies of railroads, industrialization and a Navy as abhorent to the original ideas of the CSA.
5. At least for the Lee Presidency (and that of his successor) the two parties would agree not to raise the issue of slavery and its future in the CSA but some discussion of this issue might take place on the state level.