According to FET, the sun is like a huge spotlight, and dusk and dawn is caused by it either approaching or leaving. In RET, it is caused by the rotation of the globe.
Now, the first experiment would regard the diameter of the sun. Using a good telescope and - this is very, very important - a solar filter, you should be able to see if the sun is actually decreasing in diameter at dusk, and increasing in diameter at dawn. If the sun in fact is, then it would support FET. If the sun is not, then it would disprove the current FET.
The second experiment would regard the "speed" of the sun. Since a day/night cycle is 24 hours long, the sun would have to "go faster" around the equator, or "slower" around the the north/south areas. Again, using a telescope with a solar filter, could prove if this is correct. If the sun is in fact moving faster around the equator, then it would support FET. If it is not, then it would disprove FET.
So, get your telescopes and solar filters ready. This might be a good experiment that can be repeated, unless the telescope manufacturers and solar-filter makers are in on it too...