I propose the following experiment:

• If Sigma Octantis does not move at the night time, can we assume it does not move appreciably in 24-hours (although it does move a bit with the seasons).
• Suppose a South American observer notes the compass bearing towards Sigma Octantis. Sigma Octantis is hard to find, but it should be obvious in a 15 minute long-exposure photo, as the only star that doesn't move. Once that is located by both observers, they can compare the other constellations to make sure they "match up".
• Suppose the same night (but not the exact same time) a South African observe also makes the same measurement using the same method.

Could we use this to test a monople flat-earth? If both measurements are due south and we assume Sigma Octantis never moves (has anyone ever seen it move?) then South America and South Africa can't be at the edge of a disc earth. This would provide evidence for a bipolar flat-earth or a round-earth.