The OP specified that the war goes the same way, so yes, the federal forces still take Atlanta on schedule etc. Also he specifies a "peace democrat", which McClellan definitely wasn't, so this assumes that the election happens with BOTH the frontlines where they were IOTL and the Democratic nominee is Vallandingham or Wood or someone similar.
This is one of those things that technically are not ASB, but are effectively ASB. To get this situation, I think you need a POD from before 1860. As a thought exercise, I propose that slavery is much more popular than IOTL, in fact most northern states never bother to legally abolish slavery, and the abolitionist movement is weak. The Civil War or something like it still starts due to southern bullying of the north. Maybe though legal in the northern states, slavery is not widely practiced, and they don't put much effort it returning fugitive slaves to the south, and the southern states keep pressing them to return fugitive slaves while blocking any federal government policies that might benefit the north, such as tariffs. So the nothern states elect Lincoln and the southern states leave anyway, not because slavery is under threat, more because they can't boss the north around any more. This is close enough to the actual situation (Lincoln was no abolitionist) that i'll buy it. Then Lincoln issues the Emancipation Proclamation or something like it, for reasons, and its really unpopular in the northern states, which are ITTL pro slavery, and other Lincoln administration domestic policies misfire and are badly received.
In this situation, I can see the peace democrat winning, not because the federal forces are losing the war, but because they are winning it. The voters might figure that there is not much the incoming president can do to screw up the war, since the army and navy will pretty much have it won before March, but they don't want Lincoln in charge of the reconstruction.
And the voters would be right. The only thing that might be different would be Lincoln himself sabotaging the war effort, in a fit of mental instability, by conceding everything at the February Hampton Roads conference. But if there was any sign for that, he would be removed, with Hamlin taking over until March. By the way, one problem with these threads is that, regardless of what Lincoln wanted, the Republicans were prepared to cheat to keep anything that looked like a peace Democrat out of the White House, so absent a complete collapse of the Union war effort, one would be allowed in only on the understanding that he would win the war but go easy on the South. And the latter would only happen with an unhistorically weak and unpopular abolitionist movement.