It is extremely unlikely that there is an airplane for rent in the Falklands with that kind of range.
These discussions about overflying the South Pole always crack me up. There are 2 very good reasons why there are no scheduled trans-Antarctic passenger flights:
1: There is no market for a city pair who's great circle route would take them over the continent.
2: Even if there were, the flight route would be unlikely to be approved for safety reasons. While airliners typically fly at or above 30,000 feet, if they lose cabin pressure, they would have to descend to 10,000 feet. in that event:
a: depending on how far along their route they were, the airplane might not have the range to get to a suitable emergency airfield at that lower altitude and
b: Antarctica has terrain at or above 10,000 feet, and there is no detailed topographical map of the entire continent, so terrain avoidance would be dicey.
None of that applies to trans Arctic flights, because there are plenty of marketable city pairs, there is no high terrain, and there are plenty of suitable divert fields at higher latitudes.
The problem is that you don't believe any of the hundreds of people who travelled across antarctica, risking their own lives (notably the Commonwealth trans-antarctic expedition in the 50's, and special mention to the one French woman alpinist, Laurence de la Ferrière, who travelled solo to the South Pole in 1997). According to you, even the scientists of the three permanent bases (Amundsen-Scott, Vostok and Concordia) are either liars or huge fools for people living so far inside antarctica (about 1000 kms from the shore for each), and not able to find the edge of the "icewall". I'm simply suggesting the FES to organize themselves the trip, out of the regular commercial lines, to check it once and for all.
Of course, it is extremely risky. The only airport on the direct road between Falklands and Australia is the Marambio base, on the antarctic peninsula. Out of them, there is no place to land in case of emergency. So, it will be a one-shot trial, since no rescues may come in time in the middle of the ice desert. If a drone could fly that long, it would probably be better, but we don't have it yet (maybe some day). And yes, I suppose that you won't just present yourself at the renting service, and just tell them that you are planning to check if Antarctica is not the end of the world. First, I think you will probably have to buy directly the aircraft with the appropriate fuel tank capacity; and secondly, you'll have to lie to the control tower about your destination. But then, what's the big deal since those people are already lying to you: that's only fair game. And I'm not even sure they will stop you from doing this madness if you sign all the liability waiver papers.
That's the principle of research; you can't just sit constantly waving your arguments on the net: you have to test it on the ground. Personnally, if I could prove that the Earth is flat, and a world conspiracy is hiding it, I would totally spend the money and take the risk.
And dying in space after falling at the edge of the world would be a super awesome death, don't you think?
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