The problem with this is that the KKK is not a political party, as an institution most of its (legal) activities most resemble a lobbying group. IOTL the KKK was an important part of party machines and worked mostly behind the scenes to lobby for racists legislation, occasionally doing street marches or lynchings, but most of their effort was focused on lobbying, and they were fine with that. They had no real desire to change this. In order for them to change into a political party you would need a some very unlikely events. Below I outlined what might have a possibility of making them into a political party, which could then seize power.
1928: Dem candidate Al Smith (a catholic) gets elected.
1929: Great Depression, Al Smith is blamed like OTL Hoover.
1932: The Bonus Army lead by Smedley Butler overthrows the American Government, installing a military dictatorship.
In the 1930s, Butler grows to distrust the corporations that helped propel him into power, he starts locking up large stalk holders on trumped up charges, and seizing assets without compensation, realizing that this is degrading his power base he courts the Klan to help him keep order while he purges the corporations that got him into power. The Klan agrees and the grand wizard becomes his chief of staff and Klansmen fill his cabinet. Klan legions act as security forces for the Butler regime.
Butler dies in 1940, which results in the Grand Wizard beocming head of state. For this to work the Klan would need to be filled with extremely power hungry people which wasn't the situation IOTL. Because of America's domestic instability their is no oil embargo on Japan, so no Pearl Harbor. US never enters WW2. I'd guess that this results in a SOivet Victory over Europe and a Japanese victory over Asia. This entire senrio is extremly, extremly unlike BTW, requring all of the people involved to be very diffrent on a deep psychological level then who they were IOTL.