Here's my take on it-
I don't think the issue, while Anna Anderson was alive, had near so much to do with Franziska being a peasant, and from some sources I don't think that's the reasoning now.
There really were/are a good number of people who believe that Anastasia Romanov and Anna Anderson were the same, identical human being. One's selfhood, one's identity, is a deeply precious and personal possession , maybe one of the most treasured things that any of us possess. I think the whole controversy [while AA was alive] had to do with this fundamental issue.
Let's assume, for a moment, just for the sake of discussion, that Anna Anderson really was an adult Anastasia Romanov. If that were true, just imagine what a horror her life must have been- having nearly the whole world telling you that "you" are not in fact "you."
Anastasia: Hello! I'm alive!
Relative No. 1: No, you're not. You are not you. You'll have to prove that you exist.
Anastasia: I really do exist. Do you remember when we went to the lake together? Wasn't it such fun?
Relative No. 2: You don't really remember that? Someone told you. As a matter of fact, I have witnesses who will prove that you are not really you. In fact, I think you're a Polish girl named Franziska.
Then, most of the world [listening to your relatives instead of you] denies you the right of your own identity, not for a year, or five years, but for 60 odd years of life afterward. Can you imagine what a horrible, isolated existance that would be?
Imagine having to read headlines about yourself such as "The Lady's Not a Duchess."
If AA really was Anastasia this may be the most notorious example of psychological manipulation of an individual's life [on a mass scale] that has ever occurred. Everyone had the right to decide who she was, except she. Would any of the rest of us submit to such a horrible, twisted twilight zone existance. How would you like to spend your whole life [proving that you were not dead] to a sceptical world? And by the way, who besides she really had any right to make an arrogant judgement call on her own identity?
For those who knew her and loved her [such as Peter Kurth, the Melniks, and the Schweitzers] I believe that this is the motive. It isn't that they hate or hated peasants in general It's that the whole peasant-Franziska thing was part of what they perceive as a mass injustice against someone they cared about. Part of the identity of that lady they cared about had to do with being a very educated, very cultured, very elegant [if somewhat eccentric] lady.
If AA really had been Franziska, in the opinion of these people, I don't think they would deny it [just because she was a peasant]. I think in this case they see it as a serious affront to a friend, who had a very difficult life, due to an "identity crisis."