I can help a bit, here. I think one problem is the phrase " Bryan's difficulty". It makes it sound as though the difficulty was only Bryan's. Actually, lots of supposedly very bright people circa 1890-1900 were completely convinced that the world was divided into races in competition, and that the inhabitants of someplace like South Africa were less-evolved and inferior to the supposed Anglo-Saxons who'd taken charge of it. There was also the assumption that people with disabilities would propagate those disabilities, if allowed to reproduce. A leading proponent of it had credentials: Henry Fairfield Osborne, long-time head of the American Museum of Natural History in New York.. Extraordinarily bad scientific research was done that purported to show it was all true, like the famous study of the Kallikacks by Henry Goddard. So, many people who thought themselves to be scientists, accepted Darwin, were also Social Darwinists. It was perfectly natural for Bryan to lump them together: he would die before better research was done to overturn their "findings".

Secondly, though it has been some time since I read a Bryan biography, he was not a very complex man, and he liked to think about things simply. Simplification can be an excellent tool for political campaigning, and it served him well. Likewise, his approach to Christianity was to accept the Bible at face value. Though theologians of his time were trying to come up with a response to the problems raised by Spinoza and Nietzsche, he didn't think about such things, or worry about how religion met modern science. He thought everyone should go to church , read their Bible and believe it, and be good Christians. The essential idea that humans evolved, instead of being created in God's image as stated in Genesis, was deeply repugnant to him. That meant cutting loose from scriptural authority and having to theologize, and he didn't like that, so he didn't. That unfortunately made him an easy mark for Clarence Darrow at the Scopes trial, who forced him to elucidate his beliefs. Toeing the line as to the total veracity of the Bible required him to say he believed that Jonah was swallowed by a whale and that Methusalah lived for hundreds of years, and that made him look naive and foolish- which was Darrow's intent. So, I think you could say that Darwinism didn't need to be Social Darwinism to be disliked by him, but also that he was perfectly justified in confusing the two at the time.

I am afraid I don't know anything about his dealings with the KKK, or how exactly the KKK were Social Darwinists. Though, the resurgent KKK of the 1920's was in large part anti-Catholic, and he would not seem to have been anti-Catholic himself.

A great analysis of the long, long effort by some Western scientists to validate their beliefs in their own racial superiority is Stephen Jay Gould's The Mismeasure of Man. If you have never read it, I heartily recommend it.