Your 1860 option would work, but the "somehow" is the kicker.
Its hard to think of good obvious PODs.
There were only so many Democratic politicians around who could plausibly run. Give it a year, give it a few years. But drop one off a horse, one drinks themselves to death, a few cases of pneumonia, maybe a hotel burns down, a train derailment, a heart attack, whatever. Bad luck. Spread out is better, but a couple dying at or just before the convention might be necessary. The factions of the time were ludicrously uncompromising, but they were only human; you can't rally a significant fraction of the party around a dog-catcher.
Maybe a Dem wins; maybe no one wins and the House steps in. It'd be a (proverbial this time) train-wreck, but Lincoln could be out. The next administration would be focused on another compromise.
....But honestly I'm not sure I agree with the first premise. It could easily still be a matter of kicking the can down the road.