There is an important point to make about the teaching of the church.
The question about the flatness of the earth is only one aspect of the creation of God. That being the aspects of creation which are permanent; earth, sun, moon, stars.
What we know for certain is that the Church said in the condemnation of Galileo that anyone who says that the earth is not the center of the universe and that the sun does not go around it, is suspect of heresy. Cardinal Bellarmine was of the opinion that this was de fide. (http://flatearthtrads.forumga.net/t60-pertinent-quotes-from-fathers-and-tradition
Added to this, the Fathers of the Church attacked in the strongest terms, the idea that the earth was a globe.
But we do not have clear statements that the earth part of creation is generally flat. Why? Because it is not necessary when you add up the above, (as well as opinions by Fathers that the firmament was solid). It is a way of putting it that no one bothered to do.
So Neils quest is a red herring finally. Let he who has ears listen.