For the sake of argument, I'll try to build a case for Sparkman.
He stayed out of the running in '56, and for good reason. The only event that should be changed here would be Kennedy's rather powerful campaign for the Vice Presidential nomination; however, that isn't really possible considering many politicos were pushing for his nomination long before the convention, and if Adlai were to remain as with custom in choosing a running mate, it likely would have been Kennedy.
It might be then for the best that Kennedy does win the nomination, buts gets marked by the defeat that Stevenson would suffer at the polls.
Sparkman, confident that he might be able to win against Nixon, begins building support among the Southern Delegations. Lyndon Johnson proves to be his only major rival in contesting that area, but his reluctance to leave Washington proves largely fatal to any manner of success he might have achieved. Still, Sparkman has extreme difficulty building a base of support outside the South given his signature having been placed on the Southern Manifesto, and it becomes quickly apparent that Kennedy may be the favored nominee, despite his perceived "flaws" as a national candidate.
In short, we need to eliminate him from contention.
Killing him is too cliche, and we want to avoid that.
Wisconsin seems the clinch for such a screw-up. Humphrey will be ahead in the polls but a comfortable amount given there will be some ammunition from Kennedy's campaigning for Stevenson in '56, and Kennedy will be working harder. Kennedy due to his Addison's is not able to properly provide steroids for the added stress and strain; he should know this, but let us say that he manages to slip up on this note. While campaigning in Wisconsin, a few days before the primary, he collapses from exhaustion; this has happened before in OTL. The local headlines are filled with inquiries as to Kennedy's health. Humphrey carries Wisconsin by a large margin, and Kennedy's campaign effectively ends.
I need time to think on the rest.