At the very least, the NASA ball floating in space model is utterly incompatible with the cosmology detailed by Sacred Scripture, with the firmament keeping waters from the earth. From a theological perspective, I might be willing to entertain a ball, provided that the firmament is explained ... and by explained I do not mean explained away, as referring to "space".
From a scientific perspective, I would require some other explanation other than "refraction" (which is preposterous) to explain the myriad examples of being able to see too far that have been recorded. Perhaps if someone came up with and could demonstrate a plausible theory where light bends consistently around the globe due to the electrical charge of the earth ... something like that.
It's been awhile since I've perused this topic and I'm not up to speed on any of the many extended arguments flowing out from it, but I feel like I can take a sonic ping from the other side of this argument, no matter how harsh. It's good to get a gauge on how an unpopular opinion lands.
Aside from the readily observable practical proofs of a globe Earth revolving around the Sun (the shadows as observed by the Greeks, the night sky differences from pole to pole) that had they been wildly wrong would have meant that navigators wouldn't be using great circle distance in place of rhumb lines for any significant length on an otherwise flat map, categorizing this as some sort of grave deception that feeds into a greater crisis is completely wrong and unnecessary. There is a perfectly legitimate way in which we can have a heliocentric spherical solar system without contradicting the spirit in which they were established in the Genesis narrative. If and only if it could be argued that somehow the standard model of the solar system is intrinsically and irretrievably harmful to the true Faith would I ever consider insulting someone else over it. Clearly people like Ladislaus are fine with this proposition, but I can't see how.
Some of the other stuff like the missing or severely reduced firmament could be attributed to the effects of the Flood but like Job, we weren't meant to line up the entire narrative with scientific proofs and observation either.
There is no other heavenly body than the Sun used to describe the glory of God or His power or His righteousness. The fire that consumed the sacrifice and licked up the water in the trenches came from the Heavens not the bowels of the Earth, which by the way, we are told contain Hell itself. The Earth in its comparative lowliness revolves around the Sun, as we also do (along with the rest of Creation) around the Son of God. This in my view is a perfect natural analogy to our proper orientation towards Him. The light from the Sun provides the foundation for our physical life process on Earth but even our Churches are built to anticipate the rising of it. His name is Oriens and for once our English is perfectly adapted to recognizing aspects of the Son through the figure of the Sun.
I've heard the argument that it borders on blasphemy or heresy to suggest the Earth doesn't occupy the exact center of the universe because our Lord Himself lived and died on it, but the fact that in order to appear on Earth He had to humble Himself first and submit to placing a veil over His appearance. If pride of place really took precedence concerning where he physically walked, why wouldn't he have appeared in the center of the flat Earth and some obscure distance from it? Why should the Earth so divinely touched then be dissolved in an instant and by the form of Divine fire at the end of time? It's like placing the horse before the seal.
So to sum up, I really don't see the theological need to defend a flat Earth anymore than I have to defend a square Earth with 4 corners.