Willingness has little to do with it. It would have to accept whatever terms it can get. The CSA would be in no position to push the US out of NO. If it wants it back it needs to make significant concessions elsewhere. What they would be I have no idea.
I'm of this opinion myself, needless to say. I'm just intrigued with the idea of looking at CSA scenario of the plausible, and not Moonlight and Magnolias and dashing gray uniforms. I think that the border is determinable by the land line of control, and the relative strength of any Unionist organization in the former rebel states. Tennessee, probably staying in the USA in such a situation. In Arkansas, Unionist sentiment is less strong, and in Louisiana, even less so. Thus, question marks.
Working out a plausible PoD to allow New Orleans to hold might be the start of a war-of-exhaustion scenario, though that'd be challenging with how vulnerable it would always be to naval attack, and being valuable enough to be worth constant efforts to capture.
New Orleans is hard to hold. It's so low, so close to the coast, so vulnerable to bombardment - once the forts are run, it falls. And the Union commander in question is David Farragut, never a shrinking violet or a man unwilling to fight (as men who've fought in bloody naval engagements before their voices have dropped are a special breed). On the other hand, it was famously recalcitrant.
I think, and I am now broadening to usertron's points, that to make a plausible scenario, we're pushing to the outer edges of what can be expected without giving a number of Union officers lobotomies or Confederate one's super serum.
A Realistic 191 I think Lee can have a more successful invasion of Maryland. Not Turtledove, but one where Lee moves with impunity and is not repulsed as in OTL. Lee riots around, leaves when he realizes he has no support, and McClellan stays back yammering for reinforcements as per usual. Materially, yes, Lee's invasion realistically will not do much. But this is much more damaging to the Northern psyche, and pushes back Emancipation and the hardening of that segment of Northern opinion.
Switches in Time Grant and Sherman are two of the great generals this country has produced, and were greater for their ability to coordinate and buttress each other's strengths. On the other hand, Grant drank when he was board and Sherman's mental health was fragile, especially when Grant was not around. It is easy to posit situations where they do not rise as high, or their stars do not come together.
Similarly, Albert Sydney Johnson doesn't have to catch a bullet at Shiloh. Now I'm not going to commit the sin of this board and turn A. S. Johnson into a Voltron with all the abilities of Lee, Forrest, Jackson, and Longstreet combined. But let's face it: he's certainly a better commander than Braxton Bragg, likely an improvement on Joseph Johnson.
Or George Thomas is not in the right place at the right time. Chickamauga is a disaster, and Grant and Sherman have to smash Johnson out of Chattanooga. Like the alternate Maryland above, this is not that materially different, it's just longer and more painful for the Union.
Metrics The election results of 1864 show a North that had rallied behind Lincoln. The exchange rate of the greenback to gold shows a Northern public that before that fall is a basket case. It's morale was such that even Lincoln briefly considered abandoning abolition as a warm aim (briefly), in the summer of 1864. Give it a few more bloody sieges, a few more Lee authored embarrassments, and less of a glorious record of Union triumph in the west, and it's easy to imagine this becoming dire.
So yes, the North can realistically screw up even more than OTL to the point where at some point in 1864, if Sherman or someone gets stuck in a protracted siege outside of Chattanooga or Atlanta, the North decides that it's had enough. But it takes a hell of a lot, and its a CSA that's probably not looking as it usually does in these timelines.