Theoretically, Fillmore could be elected in the following way: If in addition to Maryland (which he carried in OTL) , he also carried Louisiana, Tennessee, and Kentucky (in all three states he got at least 47.46% of the vote), then the result would be: 144 electoral votes for Buchanan, 114 for Fremont, and 38 for Fillmore.
https://en.wikipedia.org/wiki/United_States_presidential_election,_1856 So Buchanan falls short of a majority in the Electoral College, and the race goes into the House. In theory, Fillmore could win there if in addition to Whigs-Americans, he gets the support of Republicans who consider his election a lesser evil than that of Buchanan or a deadlock that will make the vice-president chosen by the Senate (i.e., Breckinridge) president.
Because the parties were in flux, it is impossible to give an exact breakdown of party strength in the House in 1856. But here is the breakdown given by Congressional Quarterly's Guide to U.S. Elections third edition):
Democrats--10 states (Alabama, Arkansas, California, Florida, Georgia, Louisiana, Mississippi, North Carolina, South Carolina, Virginia)
Republicans--7 states (Illinois, Indiana, Maine, Michigan, Ohio, Vermont, Wisconsin)
Whig--4 states (Missouri, New Jersey, New York, Pennsylvania)
(It may seem bizarre that Missouri, which had always been one of the Whigs' weakest states, elected six Whig congressmen out of seven in 1854. The explanation is the split between pro- and anti-Benton Democrats. Anyway, by 1855 most Missouri Whigs drifted into the American party. As for New York, out of 33 seats, it had 16 Whig congressmen, and 11 nominated by both Whigs and Americans.)
Americans (slave states)--3 states (Delaware, Kentucky, Maryland)
Americans (free states)--4 states (Connecticut, Massachusetts, New Hampshire, Rhode Island). All antislavery, all pro-Fremont by 1856.
Democratic-American tie--2 states (Tennessee, Texas)
Democratic-Republican tie--1 state (Iowa)
In short, Republicans or North Americans (as the Americans who came out for Fremont in 1856 were called) seem to dominate eleven delegations, Democrats ten, pro-Fillmore Americans, four (Delaware, Kentucky, Maryland, Missouri) I suspect that by 1856 even the relatively conservative Whigs/Americans of New York, New Jersey and Pennsylvania were largely for Fremont. [1] And Tennessee, Texas, and Iowa are all evenly split between Democrats and either Americans or Republicans.
My feeling, though, is that there would be enough Republicans who just could not stomach Fillmore (believing him a "doughface" no better then Buchanan) that there will be a deadlock, and VP-elect Breckinridge will become president.
If Fillmore is elected, though, one thing is certain: he will *not* approve the Lecompton Constitution. In OTL, "Millard Fillmore considered the action of the Lecompton convention "so infamous a fraud" that it would provoke the entire North to rise up against it."
https://books.google.com/books?id=Q5WF8NCK9YYC&pg=PA28
[1] FWIW, the following "Whig" or "Whig-American" congressmen from New York for the 34th Congress are listed by CQ as "Republican" candidates (successful or not) for the 35th: James S. T. Stranahan (2nd District); Guy R. Pelton (3rd); Abram Wakeman (8th); Ambrose S. Murray (10th); Edward Dodd (15th); Orsamus B. Matteson (20th); Henry Bennett (21st); Amos P. Granger (24th); Edwin B. Morgan (25th); John M. Parker (27th); William H. Kelsey (28th). Only two--Solomon Haven of the 32nd and Francis G. Edwards of the 33rd--ran for re-election as Americans (unsuccessfully in both cases). Haven was Fillmore's former law partner, and as late as 1857 he assured Fillmore that the American party had a bright future: "I think the Republican vote was accidental last fall...the causes which produced it are nearly extinct already. If our boys can hold on two years...one side or the other will come to our party." (Quoted in Tyler Anbinder, *Nativism and Slavery*, p. 247.)