While, with a less successful 1864 campeign season, the Union will be significantly more war-weary than IOTL, from a purely material POV she will still have ample resources to continue the war pretty much indefinately, at least at a low level intensity, i.e. digging in and holding conquered ground, while the South, even absent a successful Atlanta campeign and a subsequent March to the Sea, won't have the ability to dislodge Union forces from their territory in any relevant extent.
The problem isn't whether or not the Union has the material resources to continue the war. The problem is that the Union, by choosing Confederate independence as preferable to war, is no longer willing to mobilize those resources - and if I'm not mistaken, enlistments in the US army, unlike the Confederate soldiers, are not "for the war" to boot. Which means that digging in and holding conquered ground only works for - at a maximum - into ~1867 or so (possibly '68)
This is observing that recruiting is going to fall off heavily in these circumstances, so new regiments - while perfectly capable of being formed - aren't going to take their places.
You might say that the CSA will never last that long, but see below.
Even ITTL by late 1864 the Confederacy will be economically shattered, the ever tightening Union blockade and control of the Mississippi depriving it of any meaningful revenues and the possibility to raise funds to continue the war internally, just like their manpower reserves, all but exhausted, so the South will be left with little choice but to accept whatever reasonable terms the Union will offer them lest they overbid their hand and in the end lose everything by reigniting the northern fighting spirit with their from a Union POV utter intransigence.
Just as the thirteen colonies had little choice but make peace on any terms they could get?
They were economically bankrupt by the mid point of the war. They had trouble raising men (more as a matter of the former than a shortage in the sense the CSA is experiencing, but it didn't matter to the armies why they couldn't enroll soldiers). France up to 1781 was practically fighting a war that was incidental to American independence so far as its help put weight on the scales.
And yet they persevered for another four years.
Obviously the situation is not identical, but if we're talking about a peace party winning, we're looking at Virginia more or less in the same situation as 1863, Georgia comparable, and the Carolinas possibly better (due to troops being moved to make the Army of the James) than the year before.
Ignoring Florida as generally nonproductive, and focusing on this area as the Missisippi-Alabama area is able to send little and the Transmississippi is entirely cut off.
Not a good basis for "win", but too much to simply fold in another few months if the position of the armies (and thus destruction) is looking it was back in spring of 1864.
If the Union was willing to "reignite the war", it would be far simpler, less embarrassing and more effective to solve the issue of Washington's security by reabsorbing Virginia and the rest of the Confederacy than to sit on the Rappahannock, with the army's morale withering away, and hope that the CSA would rather have the mountain counties of West Virginia than northern Virginia.
So if the Union is not willing to do the former, assuming it can do the latter without difficulty just because it still has ample reserves of men, money, and supplies is missing the element of morale.