The statement that I see most often from Flat Earthers is that they are not yet certain *how* lunar eclipses happen, but that doesn't matter, because no one else really knows either. If you argue that astronomers do in fact know the cause of both lunar and solar eclipses, and that our ability to precisely predict these events is evidence of the correctness of the heliocentric Solar System and spherical Earth, then you get the Saros cycle argument: the ancients were able to predict eclipses accurately long before Copernicus and Kepler sorted out heliocentrism.
Flat Earthers also claim NASA uses the Saros cycle for making their eclipse predictions. This is based, so far as I can tell, solely on the fact that Saros is mentioned on a nasa.gov page:
https://eclipse.gsfc.nasa.gov/SEsaros/SEsaros.html While the Saros cycle is a real thing it's not how NASA or anyone else these days makes predictions.
The Saros Cycle is periodicity in the timings of eclipses. One can use it to determine approximately when the next eclipse will occur, but it won't tell you the path of a solar eclipse, nor will it give you timings for totality precise to less than a second as modern methods will.
A Flat Earther could prove their statement that the Saros is adequate for predicting eclipses by producing a computer program to predict eclipses using just the Saros Cycle or some other periodicity. No one ever has. Meanwhile, there's open source software that does predict eclipses accurately, and all of it that I've seen uses a standard heliocentric model of the Solar System and the accepted sizes of the the Earth, Sun, and Moon.