I'm sorry that this answer will be unsatisfactory and, strictly spoken, off topic, but I want to give it anyway.
The answer is: You can't. Don't waste your time. Your relative has developed a paranoia. Not only is this not an isolated earth science issue; it is not even an issue with science at all. Instead, she has dismissed our common everyday frame of reference wholesale. She must assume a lot of things along with a flat earth:
- As you say, a massive conspiracy dating back hundreds of years, involving governments as varied as absolute monarchies, Hitler's and Stalin's regimes and modern democracies. This conspiracy involves government fraud on a large scale, and a secret cooperation between all governments in the world. This renders all of our commonly accepted news meaningless, including news about economics and politics.
- No moon landing, of course. No ISS. No astronautics as we know it whatsoever. Half of NASA's employees are liars, the other half are being lied to like everybody else and duped into performing an elaborate cargo cult.
- Meteorology and climate models are based on the concept of a solar system with a near-spherical earth. Their predictions are therefore meaningless.
- Relativity, and physics in general, is wrong (light bends, the direction of gravity towards the edge of a flat earth is obviously not as it should classically be, whatever).
- If the earth is not a round body shaped by gravitation, all cosmological theories about the formation of star systems, and thus cosmology as such, must be wrong. (That also follows from her rejection of the Relativity Theory.) Then she'll have a hard time with the age of earth as well as the fossil and geological record (oh, all of geology is wrong as well!). Why don't you ask her who created the earth?
This belief of an alternate reality behind the scenes is paranoid. It also lets other conspiracy theories latch on with ease:
- Climate change is not happening, or is not happening the way we are being told, or is not happening for the reasons we are being told.
- Ask her about chem trails. They may be needed to adjust a few things. And since all commercial aircraft pilots are undercover government agents anyway lest they betray the fact that they actually do not cross the poles on certain intercontinental flights, they can as well produce chem trails on the side.
- Ask her if the lying government also keeps people intentionally sick (maybe with the chem trails!) and suppresses simple, cheap medicines.
- On a more unappetizing note, because major players in the sciences and finance system are Jews, there is an overlap with classical anti-semitic conspiracy theories (who else would secretly rule the world?).
Whatever you come up with will be countered with a new idea; you give a few examples of that pattern in your question. Occam's razor alone would do away with such complicated nonsense.1 It was sufficient to convince the catholic church, eventually.
xkcd illustrates this nicely:
A paranoid person is actually not really interested in science. A scientific answer is simply inadequate: It does not address the issue! (An example for an adequate answer is "If the earth were flat, all your horoscopes would be wrong!").
One possible psychological trigger for the paranoia is when one learns that long-held supposed truths are actually wrong. Developing scepticism as a response is in order, but the pathological response is to suddenly see all of reality differently, like in an ambiguous picture:
Let me add that there are plenty of reasons to be sceptical, because there is so much bad science out there -- most nutritional advice of the last 50 years is nonsense, for example, and most cancer screening did more harm than good. To me, that is an incentive to work towards better science; for others, it's a stumble which makes them see the world differently when they look up from it.
1 Occam's razor is actually the answer to your question, which only makes it to a footnote in my answer. Assuming roughly spherical bodies which follow gravity's pull in free fall makes a lot of observations fit very elegantly. It is difficult to come up with a *single* (or a handful of single) irrefutable proofs; defending its alternate reality, the paranoid mind is inventing counter-arguments to any single argument. But comparing the two theories *in toto* makes the judgement irrefutable. It is, in essence, the difference between the Ptolemaic and Copernican models of the world, and the jury is not out any longer: Copernicus' model is constructed "procedurally" on the basis of a few basic principles. Ptolemaeus' model, by contrast, needs many arbitrary, intricate rules for the movement of celestial bodies. Occam's razor shaves all that away.— This combining of many observations into a theory is how the scientific method works in practice. Single contradicting observations do in practice *not* immediately disqualify a theory, how ever much the mathematicians among us would prefer that. This could be seen as a generalization of Truzzi's principle: Extraordinary claims need extraordinary proof.