Considering that there is really no such thing as a flat earth, I don't think any pilot would even think of such a thing . A pilot would file a flight plan, use FAA sectional charts and would be tracked by Air Traffic Control Radar .
I have never seen anything in the way of FET navigation in the first place. No way to do it.
Interesting question but in all reality I think it is just unthinkable. Give pilots credit for some sense.
If you are real brave , just call your local FAA Office to see what they would have to say about it. Hope they listen to you and don't hang up. I worked for the FAA for 40 years.
I was also in the Navy for 4 years. The old QM's and BM's and Officers would probably be a bit more vocal in their expressions of their opinions to your question if you asked them your question. I know from personal experience that sailors can be a bit colorful in their use of the English Language .LOL
But in the eyes of the flat-earthers, the FAA and the USN are probably in the top ten ring-leaders in The Great Round Earth Conspiracy. Along with NASA, of course.LOL