One thing about getting in an argument that starts to feel scripted, like you’re performing roles, is that they’re often filling some demand other than "establishing the best argument". But if the argument is fundamentally for the purpose of winning over a third party that might work by appealing to some other interest than seeking the right answer.

So what I recommend is to pick a different role and start playing off its script, because as ridiculous as it’ll be, if you can start coherently pushing your side with the gee-whiz eager-to-explore-and-learn character of a kids’ educational TV host they will have no script for taking the other side, let alone one tuned for some secondary target.