From what I've read, a problem with the Copernican model was that it assumed circular orbits. Galileo insisted on this as well. It doesn't matter what orbits what, if you've got the shape of the orbits wrong.

Oddly enough the Catholic astronomers were willing to accept a model that had the sun going around the earth, and the planets around the sun. What settled matters for them was Newtonian physics, which provided a way to detect whether the earth was rotating or not. That was what caused them to eventually lift their ban on Copernicus. Newtonian physics also eliminated the question of whether anything in the solar system is justifiably the center of the cosmos.