This does not match my experience. As long as I "stand up" for the CS by telling the attacking or bullying civ that "they will pay," I don't lose influence with the CS. I do lose CS influence if I tell the AI that it won't affect our relationship -- i.e., if I announce that my relationship with the AI is more important than my relationship with the CS, the CS quits me with disgust (and who could blame them--I'm not even willing to protect them with words? Bah! Some protector....).