Come to think of it, isn't it a problem for a flat earth to have two different sets of constellations in the two hemispheres?No. Different stars pass over Argentina than passover Norway, but so what?
There is no southern pole star.
Perhaps you will claim that a faint star named Sigma Octantis forms some kind of an axis with the North Star.
Let's give you the benefit of the doubt and assume this is the case.
It is still the stars themselves which rotate - not the earth which is motionless.
EDIT: The classification of stars into two sets and the world into two hemispheres is an artificial classification.
Yes and no. Out in space, it doesn't exist. But here on earth.
Btw, those different stars give you a different picture than that you produced:
Who is on top and down under, of course, is an artificial determination
btw, if the Ancient Egyptians survived and wrote the maps instead of the heirs of the Greeks, this what it would look like: all Egyptian maps have the south (the source of the Nile) on top.