I suppose you could some kind of sense of the midnight sun on the North Pole within the flat earth paradigm. Because it's just a flat disc, and in the summer, the sun just circles around in a very small radius around June 21st.
However, there is another pole here and a real elephant in the room, namely the South Pole. Imagine a flat earth where Antartica surrounds the entire disc, which is the most common flat earth model theory.
There is no possible way there could be midnight sun on this continent because at any time, the sun would be farther away from the opposite side of Antartica than any other point on the disc.
So, for the flat earth theory to hold any water in this case, it has to be a lie that Antartica even has midnight sun at all, or even that days are longer in Southern Chile or Argentina.
Exactly. At the other hand, Antarctica is restricted area, one cannot go there without a permit (except Jarle Andhøy), I haven't been there myself, so I cannot personally attest that there is no conspiracy going on there.
Yes, I have not been there myself, I must admit. But it should at least be possible to ask the people residing in Patagonia if they have bright summer nights. I'd be surprised if they said no :D
Or maybe see if there is a live webcam or something, and see if it's bright from September to March. But of course, that output can manipulated :P