Seward was very strong against slavery - originally he came from the Free-Soil party, not even Whig. So, assuming as the BP that he becomes the Republican nominee, the powers who be in the party, such as Bates and Chase...
please skip the following nested parentheses if you want a summary
...(who were themselves candidates for the nomination, but they had mostly sectional support [the mechanics of the process in OTL was really everybody else trying to make sure that Seward did not get the nomination {'cause they all knew that he was really strict and probably could not carry Illinois, Indiana, Pennsylvania, and New Jersey, the 4 'border states' that the Republicans lost in 1856 <and granted that, they lined up behind Lincoln because they thought he was more likely to be a comprimise and hence win votes /and it didn't hurt matters that he had worked behind the scenes for a while before hand and got the convention moved to Chicago in his home state\ which proved to be true, largely because he took a comprimise position, which was largely because he believed in the position, which Seward could not do> recalling that Seward was hard-core, he probably could not hold this comprimise position and so would lose at least some of the 4 aforementioned states} and Bates and Chase hated Lincoln for a while afterwards, they also hated their home states for supporting them half-heartedly] and ultimately, Seward derived his strength from New York and parts of New England, not enough to take the nomination against a strong co-alition as in OTL)
...phew
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...would, out of necessity, assign the platform to be a comprimise platform. Seward, on principle, could not hold totally to this platform, and so lose a few 'border states', in this context meaning Il, In, Pa, and NJ (the ones the Republicans lost in 1856). More importantly, it woud remove such a hard-line position from the other parties. If the Democrats still split, then Douglass would not have much of a following.
More likely would be the Constitutional Union party gaining votes. Besides carrying the OTL definition border states, they would probably get some of the key Republican states.
It seems safe to say that, with only the deep south Breckenridge will not have enough electoral votes. It seems also clear that, with the upper south and a few 'border states', Bell will also not have enough. However, having lost some key states (let's say Pennsylvania and Illinois go CU, Indiana stays Republican, and New Jersey goes for Douglass as in OTL), Seward as well will not have enough electoral votes.
Then, it is a matter of politics. Essentially Douglass is now out of the running. If we arrange the other three candidates in order from south to north: Breckenrige, Bell, Seward, and say that the Republicans want a candidate as far north as possible, Seward will be out because the Republicans are smart and don't want a civil war. They will throw all of their votes in the House of Representatives behind Bell, and that combination will make Bell the 16th President.