If someone is irrational or malevolent, the only legitimate strategy is to not engage. Attempting to produce an irrefutable argument or use a strategy to bypass their dishonesty is futile and damages your own standing. Stated much more articulately by an author I admire:
"The first issue that I want to discuss is the topic of when to argue and when not--in other words, when to get into a full-fledged back-and-forth discussion, and when to refrain from it. There are many cases...in which it is not appropriate. In such cases, if it is a moral or otherwise significant issue, you could make clear that you disapprove of the person's view, so that you do not leave yourself in the position that your silence or refusal to argue implies agreement. Nevertheless, there are many cases in which, given that disclaimer by you, it simply makes no sense to engage in argument. I would summarize it like this: Assuming you want to argue--and remembering that there is no moral duty to argue--you have to decide: Is it worth it? Is it worth the strain, or do you think you will do anything for your inner clarity, or toward conversion or suggestion of a viewpoint to other people?
"Assuming that you are interested and do want to develop along these lines, I would say, in essence, argue when you believe that your opponent, however confused, is honest. He may disagree violently with you. But the question is, is he open to rational argument or not? If he is dishonest--which means that facts, arguments, and reasoning make no difference to him--then, of course, it is a waste of time. It is a waste of your breath, because you are giving arguments to someone who you believe holds that arguments are irrelevant. Moreover, if you argue with such a person, you are actually sanctioning his pretense. If he wants to engage in argument with you, that means he wants to pretend that he is a man of reason, that he has arguments for his view. If he does that while simultaneously denying reason, he is engaged in real fraud, and your refusal to comply is much more dreadful to him than any amount of refutation by you."