I understand the premise, and it makes sense, but I don't understand how you conclude that the earth must orbit the sun. Why can't a spherical sun function in FET?
It could. The observation only concludes that the sun is rotating with respect to the earth, and that the observable universe appears also to be rotating about the earth with the same period. The inference that the earth is therefore orbiting the sun is drawn from a logic akin to Occum's Razor. It is not irrefutable based on the experiment I have suggested, but of the geocentric (or flat earth) and heliocentric models it is with the least number of further assumptions that we are able to explain observable heavenly phenomena using a heliocentric, round earth model.
It is by this or an analogous method that all proofs are made. Even simple conclusions such as looking at an apple and concluding that it is thus are based on a least-assumptions model.
My argument at this point would be that if our entire world view is preposited on a certain methodology of reasoning (that which is formalised as the scientific method), we should demand overwhelming evidence to allow ourselves to diverge from such a method. FE theory is unable to provide such evidence and should therefore be disregarded on the current quality of its merits.
No, it could not if you take into account all the information we have about the Sun.
First, the sunspots are seen in the same apparent place on the Sun, no matter from where on Earth you look at them. If the FE "model" were true, the sunspots would be seen in different places from different places on Earth. Alternatively, you would see the sunspots moving all over the Sun during any given day.
Second, no matter how many bendy stupidity you want to embrace, the Sun will move from a distance of about 3000 miles from you to some 12000 miles from you every day from noon til an hour or so before dusk. Either its size, or its luminosity or both would change during the afternoon. If you bend light more, then the luminosity suffers. If you make the borders of the Sun more luminous than the center, you will see this from the Earth. No matter how you try to fix this, some other problem arises.
Finally, Occam's Razor is only useful when you have two theories that predict or explain the same phenomena. In this case the the FE "theories" do not predict a single thing right, so you cannot even use Occam's Razor. Real science wins so big that comparing both "theories" is not even possible.