It seems to me that the hypothesis of a 'flat earth' as given is easily disproved. You just need to be able to move about the planet, something you had trouble doing back in Samuel Rowbotham's day.
Simple.
Calculate the distance between cities south of the equator based on both the FE model and the RE model.
Measure the distance by actually traveling between them.
I would suggest the cities of Sydney and Adelaide in Australia. They have very close to the same latitude, which is easy to calculate in relation to the sun (And is the same in both models) but should have greatly different distances between them model to model.
On a RE, these cities appear to be about 725 miles apart. Slightly more as the road goes... but in the FE model, the distance is much more, well over 1000 miles. Easy to verify.