I rather think that Washington would have been part of the Southern effort to enforce the provisions of the Consitution. Washington would have looked upon Lincolin as a trator.
Washington would have been in a real pickle, with 75% of the Virginia electorate in favor of secession. It's an interesting speculation.
Washington the ex-president would have favored remaining in the Union, but I don't know what he would have done if offered a commission in an army tasked with slapping Virginia down, as Lee was. Washington the planter and gentleman would have gone with Virginia, I think.