Washington believed that our role as Americans trumped our role as Virginians or Nebraskans. While I agree that Lee would have acted as Washington did in 1776, it's because the reasons for the rebellion then were solid. The colonies were not represented. They did not have a say in the governments that the crown chose to impose on them. There was a situation of intolerable oppression. None of that was present in 1861. The South was represented, overrepresented, in Congress. They had literally run the government to their advantage for decades. They had their own governments. They were not overtaxed. They rebelled over petulance and slavery. Washington would not have signed on to that.
Washington would have DONE exactly as Lee did in the same circumstance. I know it, I know you know it to be true although to admit that truth deflates every reconstructed myth you have been taught about US history.