now this is just my opinion. Lee knew his opponent. Grant understood the situation. Grant would have thrown his whole army at Lee. There is no way Lee would have ever split his army. I'm not sure maybe someone can verify for me. but once Grant was actively with the Army of the Potomac, I don't believe Lee ever split his army up. Lee would have known that he was essentially on unfriendly ground. So if he lost any significant amount of troops he would known that he was on his own. If that had been Grant in charge of the Army in 1862 and not Mac and Grant had those battle plans that war would have ended at least in the Eastern theater that day. But like i said, just my opinion. Grant understood that he had numbers. Grant could lose 15,000 troops cause he had an endless supply the South did not. And Lee knew that Grant could do that and Lee couldn't.
Valley Campaign of 1864. Lee sent Jubal Early and his II Corps to defend the Shennadoah Valley against David Hunter and potentially get Grant to send some of his troops against Early. This became an independent command under Early but it was Lee splitting his forces while facing Grant in an attempt to dillute the strength of the forces he had to face.