I think the important thing would be making Grant a Southerner! I don't see how a guy from Ohio would be able to lead an army from Virginia no matter how talented he was. Lee was the axis the ANV turned on - without him they would not have been the army they were.
That said, I think Lee in command of the AoP would have been interesting to say the least. There would not have been the personality factor that was so important to the Virginians but the AoP knew a winning general when they had one.
Switching Grant for Lee is unconscionable IMHO. The Civil War is our American Iliad. Our warriors battled each other to a bitter end & along the way some became Heroes & mythological! (To the chagrin of some)
BTW fella's from Southern & Northern states did in fact fight for the other side... It was actually more common then one would expect!