I honestly think the chance of Lincoln losing the election is still pretty overrated. The election was in November of 1864 and the fall of Atlanta in early September, a full 2+ months earlier.
Sherman had only begun the campaign in May and was successfully moving towards the city of Atlanta as he outmaneuvered Johnston's army, meaning that he was achieving his objective without actually having to fight. As much as Johnston wanted to avoid a straight up fight, Sherman was just a happy to avoid one so long as he kept moving ahead. Sherman had always been a proponent of using his larger army's maneuverability as a major component of his strategy. The catch was, at some point, Sherman was going to force a battle even if Johnston had been left in charge and tried to avoid one.
For instance, Sherman could break off his drive to Atlanta and start marching around GA destroying everything in his path. If Johnston doesn't fight, Sherman would have a free hand to wreck the bulk of the state. The fact is that Johnston was outmatched on virtually every level, meaning his only real hope was a mistake by Sherman that was was severe enough to cost a major battle. Sherman simply had control of the campaign, meaning that Johnston was always going to be reactionary, and Sherman could change things up as he needed to.
Johnston could only play the maneuver, counter maneuver game so long before he had to engage. Sherman was always in the position where he was going to receive more men, more materials as the battle dragged on. Nothing Johnston was doing was going to stop Sherman in the long run; it was a delaying action. He knew this, and he was trying to pick his moment, when the battle would have been strategically in his favor. I don't believe that Johnston could have avoided a pitch battle for much longer, especially not 3+ full months until the election, and even a "win" in the short term would probably have been so costly as to effectively be a loss for the Confederacy.
In fact, Johnston committed to attacking right before he was relieved of command, as he attempted to attack one of Sherman's three columns at Peachtree Creek where he finally believed he had the advantage. Hood continued this plan, but the Union held and the Confederates were forced to fall back.
My point being is that the Fall of Atlanta was going to happen, whether Hood or Johnston held command. The Confederates were simply out manned, out supplied, Union soldiers were competent, and the Union was being commanded by a highly competent commander. Atlanta was going to fall well before the November election, Hood or no Hood.
Good reading!