A note about cognitive dissonance. If you make your point, drive it home. From what I read if you go soft after the challenge the person will grab on to any reason to hang on to their former belief, and the end result they will be stronger in their delusion. I know this sounds contrary to my usual advice. (softly softly catchee monkey) but maybe an analogy from the martial arts will help.
When the cultist side is alert to a threat, be soft and neutral in your answer. This is like the duck and weave in boxing, or the step back in martial arts. Later when you introduce disconfirming information, kindly leave no wiggle room. Drive it home. One-two-knockout.