Since it is never night in both Australia and South America at the same time, your proof is impossible to provide.
There are still discrepancies which should not occur in the RE model. For example; there are accounts of Sigma Octantis, the South Pole star, being seen South-West or South-East. This is impossible on a Globe Earth, as the South Pole Star needs to be directly Southward from every longitude..
So basically what you are saying is that this "South Celestial System" moves around the Earth to move to the other side of the Earth so that the other observers can see them at night, while at the same time this system is spinning? Wouldn't that cause the apparent movement of the stars across the night sky to not appear as a circle, but it would have to loop on it self because it is moving above the Earth as it is spinning? Sorda a tire as it is moving, if you follow 1 point on it does not complete a perfect circle because it is moving as it is rotating. And if this system is moving around like this, why does the Northern System not move either? What makes it special that it simply spins in one spot while the Southern system moves around the Earth?
Well there are a number of multiple systems which exist above our earth. The system above the North Pole; the one most of us are familiar with, rotates clockwise. The multiple systems which exists over the Southern areas of the earth rotate counter clockwise.
A person at the equator observes two opposing star rotations:
http://www.danheller.com/images/FAQ/Tech/Stars/img7.html#img10Over the Flat Earth exists a number of stellar multiple systems. Each have unique configurations and keep each other in motion via gravitational gears. Formation was caused by a conglomerate of stellar interactions during formation and the influence of the sun which makes a path through the teeth of these gears.
Here is an animation for visual effect:

The turning of the celestial "gears" is what keeps each other generally moving in opposite directions. Not literal gears, but celestial systems rubbing against each other, affecting each other gravometrically.