You asked for my thoughts on persuasion. One thing that seems important for persuasion is an appeal to more basic principles and values that you both already accept. That is, one might be able to reach a deeper principle that is inconsistent with the view one is trying to show incorrect. Let me share a personal example.
When I first read Rand's works, after a year of college, I had already become an atheist. It was already my view that only the natural world existed and that scientific investigation was how we could come to know more about nature.
At that time, I held to altruism. I held the usual view that it was wrong to be selfish and that morality consisted in rising above the natural impulse to selfishness. The best in us was to help others. (I carried out this line of conviction to politics by favoring the abolition of private property, because property allowed people to be selfish.)
There I was, open-minded, devoted to science, and subscribing to the morality of altruism. Then Rand asked me: "Why is it moral to serve the happiness of others, but not your own? If enjoyment is a value, why is it moral when experienced by others, but immoral when experienced by you? If the sensation of eating a cake is a value, why is it an immoral indulgence in your stomach, but a moral goal for you to achieve in the stomach of others?" (AS 1031)
Rand was appealing to some deeper principles of consistency and justice that we both already shared. Altruism did not really fit with those principles. So I realized it was time for me to set aside altruism as the correct morality.
I think that appealing to deeper shared principles is consistent with the outlook of your initial article "Persuasion". I know that the art of persuasion has been studied and taught since before Socrates. I'm sure there is much that is useful to learn in texts and courses on that. But I would emphasize dearly learning logic, as taught at the first-course level in college. That should include not only the formal fallacies, but the informal ones. The texts of I. Copi or D. Kelley have the right stuff. In my experience, logic is an essential for persuasion, and more importantly, one's own better vision.