And had Lee taken that offer, he’d have been a traitor to his home state and the South.
Enjoy your time at The Cemetery.
He swore an oath as an officer in the Federal Army and reneged on that oath.
He had no obligation to "the South" (Lee thought the Confederacy ridiculous and criminal until Virginia seceded) and any obligation he had to Virginia would have been to ensure that it not continue to be led by demagogues down a path to its own destruction.
Lee is a cogent example of how sentimental attachments to myths and to prejudices (and I'm not talking about racial prejudices) can cloud an otherwise fine mind.