I'd like to propose a relatively simple experiment that could be done to prove the Earth is indeed round. Enter the Coriolis Force, close relative to the centrifugal force, apparently a real force (if you believe in that sort of thing) that snipers have to account for. It causes a small but certainly measurable projectile deflection that varies depending on your latitude and the direction you're facing. Assuming a round Earth model for a moment, the force disappears entirely at the equator.

In the case of a flat and rotating Earth, the Coriolis Force at the equator should not be zero, so in order to prove (or disprove) a round Earth model, all one needs to do is find the Coriolis Force at the equator.

In the case of a flat and non-rotating Earth, the Coriolis Force shouldn't even exist (since it arises as a result of rotation), and so in order to prove a round Earth model, all one needs to do is observe the Coriolis Force at any point not along the equator.

This is easier said than done, since you have to measure the effects of a force that's 1000 times weaker than gravity, but if it's that or shoot yourself into space and take a picture, this method is clearly more feasible.