I could do with a hand in developing a conclusive test.
Lunar occultations of bright stars provide a method for determining if the stars and moon are both at the same distance from the earth's surface, as is claimed by flat earth adherents. They also provide an opportunity to calculate the distance to the moon under various assumptions of star distance, or the distance to the stars under various assumptions of lunar distance. To wit: Let us establish that observer A (in America) and observer B (in the British Isles) both locate the same bright star in the sky. If the stars are truly somewhere in the neighborhood of "about 3000 miles" above the earth, and if the moon is also "about 3000 miles" above the earth, then when the moon passes between the earth and some bright star, observer A and observer B should see that happen at very nearly the exact same time. If, on the other hand, the moon is vast orders of magnitude closer to the earth than the stars are, there should be a vast difference in time between observer A seeing the occultation and observer B seeing it.
I found a website (no doubt one of many such sites) where
occultations of known bright stars have been calculated for all of 2016. The test procedure would require you and a geographically separated collaborator to choose an occultation that will be visible from both your own location and your collaborator's location. There is a
very promising candidate on October 19, 2016 that will be observable from both Western Europe and Eastern United States, which might be suitable for the purpose. (In fact, there are four such candidates that night) The map below illustrates the locations from which the event may be observed. If one is located within the area bounded by the light blue oval to the west and the white lines north and south, you should be able to see the moon pass between earth and star, and also see the star reappear on the other side of the moon almost an hour later. Between the blue or red lines north and south and the blue oval to the east, you may or may not see the star go behind the moon, but not see its reappearance due to brightening skies of day obscuring the star.
According to the calculations posted there, the occultation will be seen from New York USA at 2:09 Universal Time, and from Bristol (closest listing I could find to Hereford) almost an hour and a half later at 3:32 Universal Time. (Notice that those times are all UT, not local time.) If those times prove accurate, it places quite a strain on the "about 3000 miles" figures, as observer B will watch the moon travel some 22° across the sky after observer A sees the occultation. In fact, observer A will see the star become visible again on the far side of the moon before it goes behind the moon from observer B's location.