You could measure this over a distance of about 100 miles, if you had observers often enough to record the height of the cloud layers, and an observer at the sunset (or sunrise) end of the cloud layers to see that the sun is shining between the layers, and an observer at the opposite end who is unable to see the sky but can see the undersides of the farthest clouds illuminated.
It would be difficult to get the exact weather conditions necessary, but possible. I suppose I don't know why a flat earth theorist would accept this any more readily than, say, the pictures of the shadow of Mt. Rainier on clouds.