I'm no fan of CSA victory in any form and so take this with a grain of salt if you like, but I think trying to break the consensus (not an unchallenged one of course, but quite predominant) that held the Union through the Civil War OTL by means of some ATL brilliant Confederate military feat is cart before the horse. If we stipulate that the Union is resolved to restore itself by breaking the rebellion, it clearly has the advantages to win in the long run; it was a question of being determined to pay the price and stick it through. OTL the butcher's bill the Union forces paid was staggering enough; we have to figure that the resolve was very strong. It took several forms; passionate abolitionism in a minority, but also the belief that if the Secession were allowed to stand than further fragmentation might cause the Union to effectively cease to exist, or the consideration that an extensive land border with a fairly populous and moderately advanced nation would put the USA in the position of European states, constantly menaced by war and locked into a cycle of militarism. A huge element of the Republican coalition was Free-Soiler western pioneers who felt they had reason to believe the Southern slave system was a threat to their own ambitions and even way of life.
This coalition would not be easy to break by some daring, bold military stroke; everyone in the North could count the resources the Union had and compare them to what the South had and conclude that even large territorial losses amounted to no more than a setback; all they had to do was carry on and they'd win eventually. At terrible cost of course; the question would be, would allowing the secessionists to have their way be worse in the long run?
So what you need, I think, is to weaken the Unionist coalition from within, and I don't think that could be done effectively if you assume that it had the character it did OTL. Basically you need to go back to the 1850s or even earlier, and lay the groundwork for the Republican coalition to be weaker, with more fault lines. In particular, you have to either get rid of Lincoln completely or make him other than he was OTL.
There are of course plenty of people OTL in modern times who find all sorts of faults with Lincoln. Certainly he had some faults and weaknesses; I think the Lincoln haters are basically living in a fantasy world with very ugly decor, but perhaps one can arrange for Lincoln to first take the lead, gathering the reins of the new party into his hands and making himself the necessary center--and then have him implode somehow or other. I gather he was subject to depression; perhaps one can make him crazier, in a way he can mask at first, just long enough to lead the Union to a precipice where everyone must count on him--then let them down by openly cracking. Perhaps he can be revealed as a true megalomanic.
I'm not sure even this would be enough to cause the Union to ultimately lose; Hannibal Hamlin might be up to leading successfully enough. I think a sick Lincoln would have to be part of a whole witches' brew of dysfunction to cause the coalition to be sufficiently eroded to lose the 1864 elections.
I personally don't care much to contemplate the consequences, nor do I think it makes for highly realistic alternate history. Perhaps it is not necessary to make every falsehood and delusion of the Lost Causers come true; perhaps quite different shortcomings can get the job done instead, but either way it is a matter of arbitrarily nerfing a historical constellation of forces that had good reasons to exist and be fairly strong and strongly woven together OTL. And redundant; it won't do to put too much weight on one figure, be it Lincoln or Grant (say, killing off Grant early)--the Republican coalition had plenty of backup. Nor can I see any realistic prospect of Confederate figures performing a whole lot better than they did OTL; my impression is that their best generals did amazingly well with what limited material they had to hand.
Insofar as glorious victories for the South are going to make a difference, I'd say they had to strike even harder (if that were possible) while the iron of the Secession was hot--earlier rather than later. Who knows just how dire the consequences for the Union would have been if Confederate forces (realistically, they'd basically have to be Virginia forces) could have seized Washington, DC before Lincoln could even be inaugurated? My guess is even that would not assure ultimate Southern victory because the Union would just create a new temporary capital, one far less vulnerable of course, and outrage would outweigh dismay. But the point is, the sooner and grander such early feats are, the more demoralizing and objectively disruptive they might be; Lincoln might come to office still determined to fight but be overruled by a public that would not stand for it. Probably not, but the South's best hope would be to act early, before the Republicans could start mustering Northern power and getting its act together. The South had no advantages good over the long haul; all their assets lay in the direction of getting a swift decision.
So that doesn't really address your scenario either, because if the Secession can secure assent, even grudging, very early on, it ceases to be the immediate issue of 1864. What you need is for the Secession to happen, the Republican-led coalition in the North to step up and attempt to suppress it, and then disintegrate spectacularly sometime after the 1862 mid-terms, so they can take all the blame.
Only that would put a Copperhead in power IMHO, and the reason for it would be because some ASB has made the Republicans rotten, some mix of inept, stupid, corrupt, or too narrow-minded to find a path to victory and disintegrating instead.