Sadly, Neil doesn't think he believes in the heliocentric system, but he in fact, does. He switched the places of sun and earth, because earth doesn't move, according to Catholic teaching and scripture...or rather, because the Robert Sungenises of the day say it is this way. Against scripture, he thinks earth is a ball. Naturally, as you already pointed out, such a silly notion doesn't work because we know that the sun, moon and stars are in the firmament, with the waters above that, and that this dome is a like a tent that covers the earth. His model cannot reflect scripture in any way, shape or form. The dome would have to be at least 93,000,000 miles away in order to house the sun! Not to mention up isn't up anymore, the horizontal horizon isn't level anymore, and water sticks to the outside of the earth ball while people walk upside down relative to each other--a notion condemned by the Church. Neil seems to think that NASA lied about the orientation of the earth, lied that it moves, but they didn't lie in claiming it is a ball. Very strange.
I wonder if one of the main reasons that some geocentrists insist that the earth is ball, is that all of the other observable planets are balls, and it would just be weird if the earth were flat, and not like all the others. And it IS a bit weird, to be sure, but then again, why shouldn't the earth be different than those other lifeless planets?
Why shouldn't our planet be different? That's what I would like to ask Neil next. Our planet has to support life, and a lot of it - from humans, to animals, birds, insects, plant life, etc.
As you said....the dome of the earth would have to be 93,000,000 miles away, in the system that Neil envisions and believes in, and that's not imaginable by any scale; it doesn't work. Though perhaps Neil (and Sungenis?) don't believe that there's such thing as a dome.
Perhaps geocentrist ball-earthers believe that Sacred Scripture is wrong - that Genesis is wrong, or that is has been wrongly interpreted. But of course we are allowed to interpret it literally. To say that it can be mortally sinful to believe in a flat earth defies logic and scripture.