If we for a moment assume that Lee does get the CO position, then it's an open question what his style of generalship is. Historically under most circumstances Lee was prone to attack aggressively to seek fleeting opportunities, but under most circumstances Lee was in a situation where his army was on a weakening trendline relative to his opponent - that is, he didn't have much prospect of his army getting stronger if he waited.
The exception to this is periods like June 1862. During this period Lee is on a strengthening trendline relative to his opponent, and he doesn't go on the attack - instead he waits while he gets stronger.
This simple fact may change the whole dynamic of how Lee's generalship appears relative to his period in command.
It's also worth really seriously contemplating the political issue, because it was effectively an open secret that victorious US generals often end up as Presidents. Careers as generals propelled or helped to propel Washington, Jackson, Harrison and Taylor to the Presidency, and Pierce served as a general as well, so effectively about one third of the presidents of the US had held general officer rank; historically of course Grant became President partly because of his career as a general, something which was then repeated after both the Spanish-American War (Roosevelt) and World War Two (Eisenhower).
If it looks likely that the Union is going to win a short war, then it looks likely that the Union is going to win a war in which slavery is retained, and Lee is really quite a pro-slavery individual as far as it goes - he's sort of antithetical to the Republican ideal. If anybody's generalship is going to get sabotaged by the Radicals one way or another to ensure that he cannot be the victorious hero who rides his military career into the White House, it is Lee. (And lest it be thought that they would not do such a thing, that sort of thing was already going on historically...)
This is likely to shape his military career, and provide a baleful influence - Davis gave Lee an almost free hand in many ways (though not all), but I do not think the same freedom would happen for Lincoln's administration and Lee.
Of course, he might win anyway.
Interesting point there that if Lee was offered and took the post, which probably meant that Virginia didn't secede, and the north won quickly then how long would slavery continue? It would still be fully legal in the loyal slave state, which is likely to include Virginia and since Lincoln had rejected treating it as the cause of the conflict it might be difficult to remove it from the occupied rebel states as well.
Its still going to happen but probably delayed a decade or so possibly.