When someone is brainwashed or otherwise influenced by propaganda, you cannot argue them out of the position or trap them in a logical contradiction. There is some literature out there on it, but they have to work there way out of it on there own. The best you hope for is to state someone that tricks someone into questioning its validity - creating a seed of doubt.

You'd probably have to state something about flat earth "ideas" that has such an obvious flaw that they can't rationalize it. But not in a confrontational way.

Also, understand that it's not about flat earth. It's filling some other psychological need or some deeper issue.

edit: I had come across https://www.culteducation.com/ a while ago and found it useful. I don't think flat-earthers are a cult, per se, but some of the same techniques could still apply.