Take one torch and one pizza base ...

It ought, one imagines, be possible to construct a model of how one believes the Flat Earth to work. Certainly one can do that for a spherical earth, and get a very good fit with observation regarding sunrise, trajectory and sunset. That way you don't have to have any fancy physics, except with a small globe all the water would fall off.

So, take your pizza base and shine your torch at it. At equinox, half the earth is in a shadow which bisects the north pole. You can ring people up and check. At the equator, the sun is overhead at noon, so logically your torch needs to be over that line on your plane (I'm assuming the North-Pole-centered model here). Is there anywhere you can place your torch such that half the disk is in shadow (ie an ant on the pizza could not see it), and it is overhead at the equatorial line somewhere? One could introduce a system of baffles, but then the ant would see the baffles, rather than what is behind them, and the sun would be above the horizon when it went behind them, which is not how we experience sunset or starlight.

An ant on the equatorial line on a spherical earth would see the sun rise straight up, pass directly overhead, then sink directly down the opposite horizon. On a flat earth it would see it moving one way in the early morning, then hang there for a while before accelerating in the opposite direction to occupy a position overhead. After noon, the morning shenanigans would be reversed. The sun would not rise and set at opposite horizons. 

Matters become even trickier to engineer at the southern hemisphere's summer solstice. One would need to allow the entire Antartctic rim to be bathed in light, but still have a special cowl that admitted starlight but not sunlight (or had the stars painted on) over half the surface. Or bendy light, whose existence certainly makes the Round Earth Conspiracy easier to sustain.

Or, I dunno, maybe spherical models work because the earth really is spherical?

The Flat Earth sun, although round in the sky (we have numerous pictures of it and it looks round), would actually have to somehow behave as a "spotlight" that is shaped like half the earth disk as discussed here - "Problem with the "spotlight sun" theory" for the model to work.

Two direct disproofs of the Flat Earth model are:

• You can not see the South Celestial Pole from above a disk, only over its edge and underneath it as discussed here - "South Celestial Pole problem".
• Travel times, by real boats, do not match the Flat Earth disk model but do match precisely the Spherical Earth model as discussed here - "Traveling around 60 degrees south".

The spotlight's shape would have to change with the seasons. At equinox there would be a dark and a light semicircle chasing each other round, with a centre of spin at the N. Pole. A couple of weeks later, the pattern would resolve into a squashed circle (due to the projection), patches of dark and light each occupying precisely half the surface area. Every 12 hours, the pattern would completely reverse - that which was dark becoming light and vice versa. So you start with a dark area surrounded by light, and end up with a light area of the same size and shape surrounded by dark 12 hours later.

During the transitions, the dark area would hit an edge, then a new dark patch would appear elsewhere to maintain the precise 50/50 light/dark pattern. The first dark patch would shrink as the other grew. All this demands some elaborate contortions from the light - to simulate what can be achieved much more simply, by shining light on a sphere, and altering the angle.

To say nothing of flying from Chile to S. Australia, which ought to be the longest journey on the planet, and take one via the North Pole. It isn't and doesn't.

It is hard to imagine what the "spotlight" would have to look like on a Flat Earth map. At equinox, it would be half the disk. At the Solstices, either the daylight half or the nighttime half would have to shrink somewhat (toward the 66.6 deg N/S). Not sure what would cause that. Yes, the Flat Earth model definitely would have problems with that.