What people here don't seem to realize is that Eisenhower himself thought that his original bill was a moderate one, almost entirely limited to voting rights. He did not understand the scope of section three:
"At first [Richard] Russell's attack sounded like the kind of apocalyptic fearmongering that one could hear across the South in the 1950s, from tobacco-stained taverns to corporate boardrooms. But Russell was not merely venting racial bile. He went on to dissect, with expert skill, the most threatening part of the bill. Section 3 was somewhat obscure, and Russell meant to shed light on it. It proposed to give the attorney general the power to appeal to a federal court for an injunction against any individual who obstructed, or who was planning to obstruct, a citizen's right to equal protection of the laws. If the injunction was then violated, and a court order ignored, a judge could assess penalties, including fines and imprisonment, without reference to a jury trial. In essence it allowed the Justice Department to use the federal courts to bypass local police forces and municipal and state authorities when a citizen's civil rights were at risk. And those civil rights were not precisely defined in the bill, leaving wide discretion to the attorney general. They could cover school integration, interstate transportation, seating in movie theaters and restaurants, and any number of fields in which the attorney general decided equal protection was being denied. Russell was right about this: Brownell later admitted in his memoirs that Section 3 'gave the attorney general direct authority to enforce Court others to desegregate public schools and to enter cases such as the Emmett Till murder.' This would have been news to Eisenhower, who understood Brownell's bill to be chiefly an augmentation of voting rights protections. Brownell had tried to slip the real import of the bill past the president and Congress.."
https://books.google.com/books?id=2PQXDwAAQBAJ&pg=PT522
Small wonder that Republican legislative leaders soon warned Ike that the sentiment in the Senate was running against section three, and that Ike pretty much acquiesced in the dropping of that section. "Senator Richard Neuberger of Oregon, an outspoken liberal, said in exasperation after Eisenhower's press conference that the president 'revealed, first, that he is not thoroughly familiar with the contents of his administration's bill, and second, that he is not enthusiastically in favor of what he does believe the bill to contain.'..Ike had thrown in the towel on Section 3.
"Johnson delivered the coup de grace a week later, allowing a vote on an amendment to cut the offending section from the bill. It passed 52-39, in what the
New York Times called 'a heavy defeat for the administration. Brownell's bold proposal to tip the balance of power from the states to the federal government had been disemboweled on the Senate floor. The federal government's ability to enforce federal law on school desegregation, racial violence, lynching, intimidation, economic retaliation, and job discrimination had been denied to the nation's chief law enforcement officer..." William I. Hitchcock,
The Age of Eisenhower: America and the World in the 1950s, p. 356. In short, there was never the slightest chance that the bill could survive a filibuster with section three. Once again, remember that thirty-three votes are all it took to defeat cloture.
Section Four, dealing with voting rights, was another matter. That Ike
did understand, and he opposed weakening it through the jury trial amendment. But even a tough section four had virtually no chance of surviving a filibuster unless LBJ backed it (and maybe not even then). And the chances that he would back it were negligible. He wanted some sort of civil rights bill to pass to burnish his own legislative credentials and to make himself acceptable to northern Democrats but at the same time he understood that the South was an indispensable basis for his getting the Democratic nomination for the presidency, and he could not afford to alienate southerners too much. But even in the unlikely event that LBJ backs a stronger section four and it passes, it is still not going to represent a revolution in race relations. Something like the 1964 bill was just not possible in 1957.