Jeremy has an excellent point.
In addition to what he's saying, there's another issue. If the sun and moon are circling above a flat earth, then their course through the sky would have some sort of curvature to it that we would most definitely notice.
And if we wouldn't notice it, then why not get a telescope and map out the coordinates to prove me wrong? Map out coordinates that prove that there is a significant amount of ellipsis in the sun's orbit above earth.
My last point: In today's modern Flat Earth theory maps, the "southern" oceans are the largest. Thus said, it would then take the absolute greatest amount of time to fly in a circle around the outermost reaches of the earth, while it would take much, much less time to fly around the "equator".
Until a Flat Earther gains control of a plane and proves this, Flat Earth theory has no proof.
Granted, the government could "shut them up" or "pay them off", but then we would come back to Jeremy's post, which cannot be argued away by the government. The only way for what Jeremy says to not be a problem would be if the earth were flat, but celestial bodies were not.
However, that would then bring other problems, such as the question of why the earth would be the only flat celestial body.