You pointed out a small date and time where you think the stars will be visible simultaneously.
It's not 'small' - it's a substantial portion of darkness, over three hours in the example used, where it is dark simultaneously in two parts of the world that are separated by roughly 90 degrees of longitude. The same is also true of South America and South Africa - I just checked on Google for today and it shows sunset tonight in Buenos Aires is at 2310 UTC, with sunrise in Johannesburg, South Africa not occurring until 0327 UTC - that's a similar overlap.
Whilst it's not easily viewed without a dark sky and ideally a telescope, Sigma Octantis is almost perfectly aligned with due south, and appears at an altitude equal to the observer's latitude in the Southern hemisphere. So on a dark night, our observer in Buenos Aires would look due south and about 35 degrees up, and our friend in Johannesburg would look south and 26 degrees up, and they would be looking at the same star, stationary in the night sky, with all the others appearing to rotate around it, for around four hours or so at the same time.
There's lots of ways of verifying that, and of course one doesn't even need to go to such extreme distances for this observation to be problematic for FET. If you just got two people a few hundred miles apart in, say Australia, to observe Sigma Octantis at the same time they would both measure it as being on the same true heading - this could not happen on the monopole FE, as our observers would be looking in slightly different directions with their backs to the north pole.
[edited to add 'slightly' in the last sentence]