Practice 30 moon landings with a simulator.
Change the gravity setting and successfully do it on the 31st try.
When Neil Armstrong did it, the final approach and landing were done "by the seat of your pants" --- there was no computer used. The de-orbit burn did use a primitive computer. The final approach and landing did not use a computer.
The craft was not flying (no air), it was translating.
There was no aerodynamic stability. - The was no aerodynamics.
When you successfully do it, post a screenshot. :)
When the trust to official entities is already ruined, that will not work well for those who in doubt. Even third-party proofs will not convince them. Say, if I visit a landing site and post selfies here, they will say it is photoshop. Or if I will not find anything and post selfies here, those who believe in landing will say I just didn't find a site. Only personal expirience could convince them, or prove that they was right.
In any case that does not really matter. Just like it does not matter if Elvis dead or alive, or did Stalin really had a pet wombat or not.
IMO it would be the ultimate test that would be easy to do.
settle it for all
It is definitely possible, but that not be at least cheap to do. So, it has to be financed by some very wealthy entity. But all that wealthy entities are highly discredited now, and whatever they get as result will be questioned.
https://www.flightgear.org/
That's wrong simulator. KSP is much more suitable. 42 is way better, but needs specific knowledge in orbital mechanic to use.
Also, if you are much into flying, it is much better to spend time and money for few real flights on some Cessna or Piper with instructor than on that computer garbage that does not give even a percent of real feelings of flight.