"Early" I understand - but later, how?
War weariness, rack up the casualties, hold Atlanta under after the Presidential election, and hope for the North to vote in Little Mac, who always over-estimated Confederate capabilities.
The PoD used in "Bring the Jubilee" (CSA gets the Round Tops at Gettysburg) wouldn't work either?
Nope. Let's assume the Confederates win at Gettysburg without taking a single casualty or expending a single round. Maybe the Union Army went to Antietam by mistake.
So what?
The Confederates don't have a siege train, nor do they have the capability of besieging the forts around Washington. They don't have supplies, and foraging off the land splits up the force, and making it very vulnerable to anything the Union forces might do between now and then. With a Confederate army, it is stone cold certain that Grant and Western troops would come east. The Union had a remarkable ability to move troops by railroad, and could get such a force into the area in combat readiness within a fortnight.
Which gives Lee 2 weeks to do something. Providing Lincoln doesn't lose his nerve, and there's no indication that was likely to happen, Lee can't take the outer forts. The Washington defences at this stage were fully manned, and Washington was probably the most fortified place on Earth at this particular moment. The next year, when the Washington troops had been distributed into the Army of the Potomac, the Washington defences were still under no threat when Early arrived and failed to win at Fort Stevens.
Lee's got no siege train, no time, no supply lines, no way of staying still and no way of moving forward. There's no hope of Britain recognising the Confederacy at this stage - not with the Emancipation Proclamation out there.
Gettysburg is the single worst PoD to get a Confederate success in the ACW.
Mind you, Bring The Jubilee handwaves the Confederacy ending slavery because of the efforts of Lee. Lee, who couldn't even free his own slaves when he was instructed to do so by his father's will, and who took free blacks into servitude during the Gettysburg campaign, is - so we are led to believe - going to overturn the slavery that was specifically written into the Confederate constitution. John Mosby was certainly in no doubt, stating that in later years, it had become obvious that slavery was a monstrous evil, but that at the time, that is what people were fighting for.
Bring the Jubilee is a classic of the Lost Cause romantic nonsense one sees.