For what it's worth, the FE hypothesis is that the stars are merely a few thousand km from the earth. That means that two observers on Earth separated by about 5,000 km can simultaneously take a picture of the
Moon and the parallax should show the Moon appear shifted by 13.8 arcmin (78.5% of the FE Moon's angular width) as compared to the background stars (assuming the stars are right behind the Moon). The actual apparent shift of the Moon that we should observe empirically is about 44.7 arcmin (143.9% of the observed angular diameter).
This observer separation is approximately the distance across the US and the experiment could be easily performed. I hope to actually perform this experiment.
Interactive diagram
here.
To answer your question though, my guess is they would make up a pure speculation about refraction or something. All without
any prior empirical evidence or models to back it up.