I seriously doubt it. The Union, after fumbling around for two years (arguably in part due to the desertion of quality officers to the Confederacy), had finally solidified as a cohesive military group, and after Lee’s move on DC, I doubt they would have been interested in much quarter for the Rebs.
As you mention, the Union had the Mississippi, effectively surrounding the Confederate states - blockades on their ports in the east and south, the North itself, and the domination of the west. Kind of a tricky position to get oneself out of - surrounded strategically, cut off from most hope of supply, most able bodied men already volunteered, in retreat, and with a superior force marching down your throat. The Europeans weren’t going to get directly involved, whatever the South may have wished. Keep in mind, there was some (justified) fear that America would soon have a mobilized army of battle-hardened veterans, and Canada was sitting up there innocently trapping beaver and trying to build their government (and fend off the Fenians, heh), and I doubt the Brits would have risked another war on the American continent. The French were making their play for Mexico, which was risky enough. Plus, they were still eyeing each other - if anyone made a move to ally with the Confederates, you bet your sweet bottom that another power would offer their aid to the Union (who, at that point, would finally be viewed as the likely victors).
In any case, I don’t think the Rebs were terribly interested in negotiating their surrender in 1863, hope or no hope.
So, the question is - could the South have successfully retreated, regrouped, resupplied, and mounted another offensive? Only if the Union did another “golly shucks, lets just sit here and let them dictate the war”… not bloody likely. They were already mobilizing to go on the offensive - if not through Virginia, then through the deep south. Either way, the Confederates would be fighting a two front war which they really weren’t prepared for - as I recall, their plan was to capture DC and make it a quick fight while securing their own borders, raids into the midwest notwithstanding.
If the Union hadn’t been striking from the Miss, the South could have likely retreated far enough to regroup - but would probably never have the penetrating power they started with - however, since your question is post-Vicksburg, that scenario is out the window. With the Union pressing there, I don’t see a way for them to turn it about.
Though the rise of the South into a full fledged nation, almost gaining recognition by France and England, is fascinating, I personally think they were doomed from their first failure to take DC. In any prolonged conflict - well, they were an agricultural, aristocratic institution facing a industrial democracy. Their whole chance was taking the capitol out quickly and forcing a truce. Once they failed in that, they were doomed. By the time 1863 rolled around, the Union had rebuilt its military, mobilized its industry, polarized the conflict by making it about slavery (effectively pinning the Brits and French to the wall), repelled the biggest thrust, and opened a second front, the war was decided (as well as the lives of hundreds of thousands of people)
(All of the above is what I remember from high school, I never studied American history in college 