That's pretty hard to imagine. Lee has been lauded for his loyalty so for him to betray his country and his military oath of office, just because some in his state and some neighbors felt that the president's politics MIGHT, EVENTUALLY threaten their businesses that used slaves, seems out of character. It's hard to imagine him deciding that the zip code of his state is more important that his country.
Additionally, his plantation sits RIGHT next to DC so he knows that's going to be a HUGE target. If he joins the CSA who knows what will become of his home -- it could become a graveyard for all we know.
In any case, if Lee somehow does go crazy and join the CSA, it could make a significant difference. He is a respected military leader and could be an important player in their military. It's not clear how far he'd go in their ranks, be he'd start with seniority to all but a few. I suspect that Cooper or Johnston would likely start out in command and there would be several other contenders. Some might even feel that Lee was a little too old for field command and have him be Davis' military advisor, or some such. If Lee is used in the field, he might reach Corps command, possibly head some small region, or possibly be chief of Confederate artillery. I find it hard to see him ever reaching overall command because he wasn't a favorite of Davis and his aggressive generalship wasn't ideally suited to the type of defensive war the CSA HAD to fight to have any chance of winning. In the Union army that style was appropriate most of the time and he knew how to use the troops and firepower he was supplied with. He was a natural choice for command after Lincoln got sick of McClellan, Hooker, Burnside, etc.
As for Grant, his skills were going to allow him to rise to high leadership regardless. But, if Lee hadn't served able as commander of the AOTP from 1863 on, Grant might have been given that job. Instead, it made sense to have Grant lead all US forces outside the DC-Virginia area and leave Lee to fight the main foe.
Finally, if Lee had joined the CSA, the war might not have ended in late 1863 as he'd help the South and he'd be missed in the North.