This would require the use of an aircraft, although that should not be an overly difficult problem for a dedicated researcher. First choose a departure point and an arrival point which both lie along the same approximate latitude. Now since on a spherical world the line of latitude would be straight (apart from the curve of the sphere, obviously) then it should provide the shortest possible route. However on a flat earth a straight line between the two locations would deviate from the latitude line. The test is this: fly between the two locations twice (there and back each time so you have more data to work with and you still end up near home), the first time follow the line of latitude, the second time use the straight line. The straight line should be easy to determine by drawing it on a two dimensional map showing the curve of the latitude line and following the line using the locations the line crosses. This would obviously be easier done over land but can also be done using GPS points over water. There should be a discernable difference in the times each flight takes, if the "straight" line is shorter then FE wins, if the latitude line is shorter then RE wins.
Would this work? Obviously keeping all else equal such as airspeed and altitude, etc. FE and RE opinions please and if it is considered an unreasonable test please provide reasons for this, I'm not a pilot so I won't know unless you tell me.