Mike, his own party just voted it down prior to his inauguration.
But the Republicans in Congress voted it down only because Lincoln told them to do so. If Lincoln, as president-elect, had told them to support it, they would have supported it. We know that a number of them wanted to support it, but Lincoln was adamant and they didn't want to buck party discipline.
It would be true that had the senators of seceding states voted with the minority who voted for it, it would have passed. But they didn't.
They didn't because they knew that for the CC to be the most credible and enduring it had to receive bipartisan support, as had all other similar compromises. A "compromise" passed on a purely sectional vote could be much more easily and quickly undone by the next president, who was Lincoln. Lincoln once again was the key. The Southern senators knew that if they passed the CC on a purely sectional basis, it could end up being worthless if Lincoln refused to support it.
Further, Lincoln's platform was "no new slave territory."
But that's just the point: The CC would have ended slavery in the few areas of the territories where it then existed and would have allowed it only in the region where it was already clear that slavery would never take hold. This was simply a bogus issue, as historians like Albert Kirwan have shown at great length. It was irresponsible for the Republicans to keep stirring up passions over it when it was already very clear that slavery was not going to ever take hold in the territories.
He was elected with a minority, as you say. Imagine the political carnage he'd have incurred amongst his own supporters if he told them they'd won an election but now he was abandoning the central issue of his candidacy.
One, that would have been far, far better than the bloody, horrific war that followed. Two, that's what true statesman do when necessary--they buck their own party's desires. Three, the situation had changed radically from just a few months earlier when Lincoln was elected and now quite a few Republicans, including Republican newspapers, were calling for accepting the CC.
It was mainly only the Radical Republicans who were ardently against it. So the political carnage would have been minimal because the CC was extremely popular, which was why the Radicals didn't even want to allow a national, non-binding referendum on it--they knew the pressure to pass it on a bipartisan basis would be irresistible if it received a large majority in a national referendum.
The compromise might have been possible in 1858 or so if a national and unifying leader had pushed it.
The CC was entirely possible in 1861 and in fact was wildly, overwhelmingly supported by newspapers across the spectrum and in petitions that were pouring into the capital by the thousands (many of them with hundreds and even thousands of signatures, which was quite unheard of until that time). Even Greeley admitted that if put to a popular vote, the CC would pass by a wide margin.
The problem was that Lincoln refused to support it, even though it favored the North on every slavery-related issue, even though it banned slavery from the few territorial areas where it (barely) existed, and even though the 25% of the territories where it permitted slavery was a region that clearly was never going to be plausible for slavery to take hold.
Lincoln's senseless, stubborn refusal to compromise was a tragedy that paved the way for a horrible war that not only killed nearly 700,000 Americans and wounded millions but also caused bitterness and strife for decades after it ended.
If we had had a true statesman like Clay or Webster or Crittenden, or even if we'd had Seward or Douglas, war would have been avoided, secession would have ended and collapsed, the Union would have been saved, and slavery would have died a natural death in a matter of a few decades. Let's keep in mind that it took 40 years for the free states to abolish slavery--from the time the first free state passed an emancipation law until the time the last slave in the free states was freed was over 40 years, and the world didn't end because emancipation was so gradual.