The Klan would have to be the "power behind the throne" (rather than having the President and a bunch of congressmen burning crosses on the White House lawn or something). In that no one could run for/win an election without Klan approval. This isn't too far out of the question, since well into the 20th century in many places, you needed to have the approval of the local political machine to win even the primary (which was in places like the South, the real election). So what we're presuming is that the KKK takes over/becomes the political machines that dominated local/state politics.
And that's pretty hard, since the Klan seems hard to make compromises with. Apparently, they had an auxiliary group on the same status as the Women's Ku Klux Klan meant for blacks, as well as one meant for naturalised Protestant citizens. How can these groups be any stronger, when that would compromise so much of what the Klan was and what gave them their appeal? And for that reason, they can't take over/defeat the political machines on a large enough basis. These are the sort of machines which helped black people in the South vote during Jim Crow whenever it suited their needs. You're gonna need a very different Klan to defeat the strength, and with such an alienation of blacks, Catholics, and immigrants in general, they simply can't win.
Nationally, I still doubt it. The Klan can sway major elections their way. But their enemies are still the local/state machines, which can sway them the other way. The Klan would have a difficult time of swaying presidential elections more than they did OTL.
But less Catholics and immigrants in general in the US might be able to help the Klan out. However, would you even have the Klan have so much strength at that point? Anti-black racism would only take you so far in the 1920s in the places the Klan's powerbase was (Indiana, etc.), simply because of how few blacks there were there at that time.