What experiments could one do in order to distinguish FE from RE?
Larger/longer versions of the Bedford Level.

Rowbottom asserts he was looking straight and level along the tops of the flags. If the height of the flags is 5 feet, Rowbottom's eyes and/or optical instrument were also at 5 feet.
If Rowbottom had a straight, unwavering sightline over the various flags, as the illustration suggests, then we can assert a straight sightline from his eye/optical instrument down to the surface of the canal, or to the base of any the flags, at zero feet above the canal. Why not?
So;
Find a spot onshore, looking outward/seaward toward boats/ships or islands of lesser height than the observer. Let's say observer height is 100m. Sea level is zero.
Let's presume the sea to be flat, as per Rowbottom's assertion for the canal.
All straight sightlines to the sea's surface must be downward sightlines. From 100 down to 0. They cannot be upward or level. The observer cannot be looking upward or level toward the sea below him.
All sightlines must pass through all levels below 100m on their way to the sea. 100 down through 95, 90, 80, 70, 50, 25, down to zero. Cannot pass through 110, 120, or any other level above 100.
By the same token, if the observer sees any object out on the water of lesser height than his, say at 80 or 25, the sightline through the top of it must, if continued beyond the top of that object, meet the water.
The downward sightline is non-parallel to the plane of the sea, and must always meet the sea at some point. Non-parallel lines always meet.
If the observer does not see water behind and beyond the top of the lower object, the seas CANNOT be flat. If there is clear sky behind and beyond the top, the seas CANNOT be flat.
Simple right-angle triangle geometry dictates the point at which the straight sightline must meet the water, where the observer height, object height and distance between are known.
The following observations are therefore an impossibility if the seas are flat; the first is my own, the second taken from a flat-earther's YouTube video.
Observer height 100m, ship height 52m (the one on the right with two cranes);

Observer height 210m, lighthouse on the island 73m;

Nothing to do with "ships going over the horizon", "disappearing bottom up", etc etc ...