As a relevant skill in your situation, you should learn to distinguish refuting and rejecting an argument. Rejecting an argument refers to not changing your own state of belief, while refuting an argument refers to changing other person's state of belief. If your response doesn't change another person's mind, then you've merely rejected the argument, not refuted it. (Counting some activity that doesn't result in the other person's change of mind as refuting an argument defies the purpose of the terminological distinction.)

If we are talking about an old-time cult member, refuting their wrong cult-generated statements is nearly impossible, so usually there is little point in arguing at all. The goal should be to perhaps minimize conflict, but not to convince, unless you actually expect a nontrivial probability of success, which you should recognize as usually absent. Don't play the lottery.