The guerilla war in Missouri existed BEFORE the American Civil War. But it did so because the Southern Democratic Presidents sitting in the White House at the time were too sympathetic to do anything about them. With Abraham Lincoln at the helm, and the rest of the South crushed, the Union Army is free to do with them as they wish.
The Appalachian region were it was thought a guerilla war might also be fought were too Unionist to support Confederate resistance fighters.
And assuming Lincoln is still assassinated, any guerillas attempting to resist will see a grand scale urban renewal response by the Union Army. The SW third of Missouri, where the Confederate resistance movement was concentrated, was relatively sparse on the ground compared to the rest of the state. You could see that region being completely depopulated by this.
The idea of a "Peace Democrat" being elected, much less Lincoln's losing forcing an end to the Union war effort, is a myth. Usually put out by people who don't understand the meaning of American political party platforms going into an election plus ignoring the nature of George B. McClellan himself. The Democratic Party in 1864 coming out of its Convention could have had a platform calling for the Union's Unconditional Surrender to the Confederacy

, but that wouldn't mean that President George B. McClellan would in any way be bound by it.
The Congress DECLARES war, but it doesn't END them. The President is the one who NEGOTIATES a treaty of peace, surrender, alliance, or cease-fire. Congress merely approves or rejects it.
McClellan was a WAR Democrat, not a PEACE Democrat. And he ran as such. Something that those who treat the idea of McClellan winning = South winning have no end of trouble understanding. McClellan fully intended on snatching the vittles of victory from Lincoln's plate and taking credit himself. After all, no matter how much the Peace Democrats in Congress may fume, there will still be enough War Democrats and Republicans left to form a working coalition to run the Congress, even as Sherman and Grant break through in Georgia and Virginia in 1865, once the terrain dries.
The Copperheads could never hope in these circumstances to defy McClellan, short of impeaching him, which would require the 1866 election to be a Copperhead landslide. Incredibly unlikely, as the CSA will have been overrun by this time. Otherwise, the Copperheads simply won't have the votes.
Biggest change will be that the Union will not pass the 13th, 14th, and 15th Amendments.