First: Whether or not the CSA can accomplish anything after it is independent, or whether it can even avoid disintegrating within a few years/months of its independence, is besides the point. If the US recognizes the CS as independent, then the south has "won" in the strict sense of the word.
As for getting that to happen, it is simple enough! Come on people - yes the south has to be lucky, but not implausibly so. In fact, my preferred TL for this remains one with a BP *after* Gettysburg/Vicksburg. But in general, just let the Confederacy hold onto Atlanta, and for good measure, the Shenandoah Valley and Mobile, going into the 1864 elections, and McClellan wins on a peace platform. If they can then hold Atlanta, at least, until McClellan is actually inaugurated, they are strong enough to get a brokered peace.
Getting this to happen is simple enough:
Bragg gets sick in August 1863. With no other viable replacement, Lee is sent west and takes over the Army of Tennessee. Chickamauga is an even bigger Confederate victory, with them capturing >10,000 Union troops including then-division commander Sheridan. Lee follows up by leaving just enough of his army in place at Chattanooga to hold the Union there, then sends ~3x OTL strength to Knoxville to re-open the railroad, upon which most of the force returns to Chattanooga in time to face Grant. Due in part to well-placed entrenchments, and in part to Lee keeping a division in reserve, he holds the line on Missionary Ridge, counterattacks, and captures Sherman and ~20,000 men of the Army of Tennessee in a virtual peninsula on the Tennessee River. Now significantly outnumbering northern forces, Lee sends a corps west and then north along the railroads in Grant's rear, threatening to cut his only supply line, and forcing his retreat back through Middle Tennessee - one more battle near the old Stone's River battlefield lets Grant get away cleanly, but the Union army is bottled back up in Nashville.
In 1864, Lee travels back to Virginia, facing newly-promoted Lieutenant General Thomas (the only person in the west who consistently did a credible job) - with Grant elevated to Sherman's OTL position, but all the way back in Nashville, and facing Johnson. Thomas tries a slow, methodical, campaign of maneuvre in Virginia. Lee defends expertly but is slowly forced back - both sides avoid pitched battle, at least where there are earthworks, so casualties are low. In Tennessee, Grant half-wrecks his army attacking Johnson's entrenchments, but ultimately he wises up and begins flanking maneuvres as well. Johnson falls back a bit faster than Lee does, but by the 1864 elections is still holding Chattanooga. With no appreciable gains anywhere else (no Sheridan = no Valley), Lincoln loses in a close election. McClellan, seeing that there are no victories anywhere, is persuaded to stick to the peace wing of the Democratic Party. *Maybe* over the winter, Grant takes Chattanooga, but Johnson then has the mountains to defend Atlanta, which he still holds on March 4th. Thomas tries one pitched battle in front of Richmond, and after the bloodbath (for both sides) concludes, it is close, but Lee still holds the Confederate capitol. McClellan immediately engages Davis in peace negotations.
Of course, if you want a victory even earlier than this, then starting with TL-191 as a base may be helpful, if you do it smartly. Yes, there is no Antietam and Lee keeps moving north, but you need something more than that - as in, get rid of Bragg as fast as possible. Hardee, say, or even Smith, vs. Buell in Tennessee leads to a Confederate victory over a *wing* of the Army of the Ohio. Said army is too large to take off the map, but having a strong Confederate presence in Kentucky at the same time that Lee consolidates his forces in central Maryland/southern Pennsylvania may even be enough to get Britain to intervene.
Or, another option: let Bragg be lucky for once, and at Stone's River, Breckenridge is smart/not-feud-ful and sends most of his division west of the river in time to support Hardee's attack. He cuts Rosecrans' supply lines back to Nashville, and ultimately outflanks him. Even if Bragg cannot destroy the Army of the Cumberland, he can force it to retreat back to Nashville. *That*. coming on top of Fredericksburg, Grant's retreat, and Chickasaw Bayou, (and on the same day as the E.P.) leads to huge riots all over the north, way bigger than OTL draft riots. Copperheads even get the moral high ground in places - I'm sure all of your creative minds can butterfly this into a Confederate victory.