Even by forcing the election into the House however, the Democratic Party at the time held the majority of State delegations and would have been able to elect Buchanan.
I don't know the source of your map, but as I indicated at
https://www.alternatehistory.com/fo...buchanan-fusion-in-1856.349796/#post-10572481
here is the breakdown of the 34th Congress delegations 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 seem to dominate eleven delegations, Democrats eleven, 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. A quick glance at
https://en.wikipedia.org/wiki/34th_United_States_Congress seems to confirm these figures.
Maybe your map comes from using the party breakdown for the 35th Congress (1857-59)? But before the 20th Amendment, it was the old Congress that decided. (In fact, not even all the new Congress would be
elected by the time the House would decide on the new president; there was no uniform election day for the House in those days.)
Anyway, to get back to the 34th Congress: My own guess is that if the election goes to the House, there is a deadlock leading to Breckinridge being chosen acting president by the Senate. To many Republicans, Fillmore was a "doughface" in no way preferable to Buchanan; and this is true not only of those Republicans who had never been Know Nothings, but of some who had been. OTOH, unless the "South Americans" massively desert Fillmore for Buchanan, I don't see Buchanan winning, either. (To be sure, South Americans would vote Democratic to defeat a Republican, as was shown by their overwhelming support for Aiken in the Speakership fight against Banks. But the disproportionate power of the South under the "one delegation, one vote" rule guaranteed that Fremont could not win in the House. So there was no need for South Americans to support Buchanan--when the worst their opposition to him could do would be to put their fellow southerner Breckinridge into the White House.)
[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.)