Oh, all that was obvious to me when I wrote the text. I’m sorry I could not convey the logic of their actions better. I don’t think that Harry endangered Snape’s position, because either the Shiftings believed Snape was on their side and had duped Harry and therefore expected Harry to behave the way he did, or they didn’t believe Snape was one of them even to begin with, and then it did not matter what Harry said anyway. Harry’s behaviour does not seem to harm Snape - the fact that Snape is contradicting and humiliating Harry only adds to his credibility, until he is asked to cast the killing curse on Frankiss, which Snape cannot bring himself to do. Harry, more than Snape himself, has already realised that Snape is incapable of going back to the position as a spy (he told Snape so already in chapter 41) and does not think that playing along is a solution that will get them anywhere. 

Remember that in this story, Snape was mainly used by Voldemort during the war for collecting intelligence. The brutality of his days as a ’real’ Death Eater lies far back in time and even then he had difficulties killing (described in chapter 32 in parallell to Harry’s adverse reaction to the Avada Kedavra in Paris). 

Seriously considering to force himself to kill Healer Frankiss fundamentally affects Snape. Until then, he actually believed he still had it in him. I’ve tried to show to what point the outcome of Snape’s trial shook him. He obviously thought that the only solution would be to be sentenced to Azcaban, let himself be rescued by the Shiftings and resume his role as a spy. Now faced with the reality of the fact that he cannot play this role once again, he deflates completely - thence the sense of defeat, thence the freezing reaction.