This is back after almost two years?
I don’t find the apparent discrepancies in Harding’s actions all that hard to believe.
Harding was a personally compassionate person, but was not a deep thinker and was rather irresolute. He, himself, noted that he would listen to his advisors argue one policy and agree wholeheartedly, then hear a different set of advisors argue the opposite and find himself immediately changing position.
The original Klan only lasted from 1866 to about 1869 (some factions until 1871). In 1915, inspired by the movie The Clansman (renamed Birth of a Nation, Wlliam Simmons recreated the Klan. The nativist and racist movements that were swelling up in the late teens helped to bring the Klan to unparalleled membership, with their national publicity emphasizing their Americanism rather than the hatred and the anti-black, anti-Jewish, anti-Catholic, and anti-immigrant actions that were perpetrated on the local levels.
I would not be at all surprised to discover that Harding accepted an honorary membership in the Klan (played out in a publicity stunt in the Oval Office or Green Room), based on a notion that it was simply a good thing to be associated with this rising “American” movement.
(I would also not be too surprised to discover that the event was a fictional claim by Klan supporters and never really happened. Looking over the conflicting claims that are available–that Harding might have had black ancestry, that he definitely had black ancestors on both his mother’s and father’s side and knew it, that he was a Klansman in his youth (at a time when they were actually disbanded!), that he joined the Klan before he was elected president, that he was inducted into the Klan in the Whitehouse (alternatively in either the Oval Office or Green Room), etc.–I would need to see rather specific documentation before I believed any of those stories.)
Regarding the OP: Jomo Mojo’s link is now dead, but it is repeated at http://www.mdcbowen.org/p2/sf/faq068.htm
I find its information suggestive, but, unlike its author, I do not find it persuasive. Without hard evidence, I suspect this will be debated for years.
As to the actual bolded line in the OP, I have no trouble believing that Beethoven rejected the appelation “German”, but I would have to see better evidence than a claim by Garret Morris that Beethoven ever objected to being called “white.”