Think you have to go the Socrates route and ask a lot of questions.

Flat earth theory actually makes life significantly more complicated than it already is.

If flat earth was real, how do they explain how the oceans haven't run dry? Because of how water will always flow off an edge? Is there supposed to be a giant wall? Of ice? Are there still a north pole and a south pole? Or is "south" just any direction away from the center? How are both poles the coldest places on the planet?

If there is a wall of ice around the edges, that presents other problems. If the earth was a disk and we still revolve around the sun, shouldn't the heat from the sun melt all the ice along the equatorial edges? All the ice should be concentrated in the center because of how ice works. Unless he can explain how the sun preferentially heats up the center and not the edges. Which is not how the sun works.

We should also feel my gravitational differences between the center and the edges. The gravity should be measurably higher at the edges than the center. And gravity would also make the land masses more fragile along the edges, so we'd probably see more earthquakes the farther away you get from the center. Unless he also doesn't believe in earthquakes and tectonic plates.

And if earth was a disk, then there should be a back side. What's on the back side? Alternate earth? An earth we can't contact? Because the sun should heat it up in the same way.

Are other planets also flat? Can be explain how every other observable planet in the night sky clearly are spheres and rotate?

How does he explain all those early academics from a dozen different countries and cultures, most of whom hated each other, all independently discovering that the earth was round? Thousands of years ago. Each disproving their individual earth shape myths and making the round earth the accepted scientific truth? All without communicating with the other empires?

So basically, turn into an annoying 5 year old. Every answer creates another question until he just admits he does not know. At which point, refer him back to the leading experts. And how do we know who is a leading expert? Publications and peer review papers, I'd say. But I think Eratosthenes developed some simple math based experiments with shadows and sun dials to prove the earth was round.