This is the core problem of scientific communication. Flat earthers are an extreme example. The process of arguing it etc can be a good exercise.

Edit: my favourite argument is to look at a half moon, and what angle the "shadow line" on the moon is relative to the horizon. Then ask your friends around the globe to report what angle it looks like to them. Because we are all standing on the side of a globe, we see it at different angles relative to our local horizon, which should perfectly correspond to each person's latitude etc. Fun easy experiment for an online community!

Yes, but the round earther can provide a model that matches your observations. Round earth is a better model because it matches reality more often while also accounting for a lot of the obvious flat earth arguments (mostly, that they hold true locally).

But they don't care. And more importantly, don't want to agree with you.

Then it's not persuasion in good faith.

Of course. Good faith is pretty rare and subject to, shall we say, hostile evolutionary pressure in online and public policy spaces.

Go to the shore. Watch the ships. It's not complicated sciencey stuff.

Atmospheric refraction. (See? The flat earther knows complicated sciencey stuff too. You won't accept their complicated sciencey stuff, so why should they accept your complicated sciencey stuff?)

We’re not trying to come up with a convincing argument - there’s thousands - but we’re trying to understand how it’s even possible to conceive of such idiocy for a person who is otherwise reasonable.

That one is easy because it’s directly refuted by lived experience — eg:

There’s mountains outside my window, so… not flat.

Do you have evidence that it’s flat? — what do you even mean by “flat” when I can see ripples in it?

They mean “flat” in the sense of “straight” (i.e. not curved), the same way a sheet of sandpaper is flat despite being rugged. There are plenty of simple ways to disprove Flat Earth theories, but you’ll never be able to convince anyone by refuting their arguments with something which clearly misunderstands what they’re trying to say.

Take the piece of sandpaper and put a twist into it and now fold it back upon itself. Now it's not flat…nor is it round. It's a secret third thing.

Anyway fun topology aside a lot of flat earth discussion is not really done with the attempt to disprove anything, unfortunately.

Instead of refuting their arguments, ask them to prove it to you. Whenever they say something you don't understand, ask them to explain it. Eventually they will get stuck.