I've never been to Australia, don't know if there are any difficulties about shooting dingos in the dark, but, simply, "prove the southern stars exist" -- well I can't even prove the northern stars exist, except as points of light in the sky. I suppose astronomers and astronauts, using telescopes and high tech equipment such as radio telescopes and satellites, could prove they exist - IF FEers are willing to accept the high-tech equipment. Anyway, what they can do with the northern stars applies equally to the southern stars.
If you use a star map, such as appears (sometimes) in a world atlas or an astronomy textbook, you can see the stars and constellations of the southern hemisphere. But the southernmost constellations have "modern" names and configurations, such as "the microscope" -- applied in the 19th and 20th centuries by Europeans who were utterly indifferent to what South American Indians and Africans and Australian Aborigines might have called the same stars and constellations. They can be seen if you go south of the Equator and the further south you go the better view above the southern horizon of those stars never seen north of the Equator. And, yes, they do circle around the sky as the hours pass, centered on a spot not occupied by a star but calculated to be due south.
It seems to me that, if the earth is flat, it's tough enough to explain why the stars spin around the north pole without trying to explain why they also spin around the south pole - if the earth is flat. I would posit the cycling of stars around both poles simultaneously as evidence of a round planet.