I have taken the stance to not argue with LLMs, don't give them any clues, and don't ask them to roleplay. Tell them no more than what they need to know.
And if they get the answer wrong, don't try to correct them or guide them, there is a high chance they don't have the answer and what follow will be hallucinations. You can ask for details, but don't try to go against it, it will just assume you are right (even if you are not) and hallucinate around that. Keep what you already know to yourself.
As for the "you are an expert" prompts, it will mostly just make the LLM speak more authoritatively, but it doesn't mean it will be more correct. My strategy is now to give the LLM as much freedom as it can get, it may not be the best way to extract all the knowledge it has, but it helps spot hallucinations.
You can argue with actual people, if both of you are open enough, something greater make come out of it, but if not, it is useless, and with LLMs it is always useless, they are pretrained, they won't get better in the future because that little conversation sparked their interest. And on your side, you will just have your own points rephrased and sent back to you, and that will just put you deeper in your own bubble.