I'm the first to admit I don't know a lot about the beliefs of modern day 'Flat Earth' believers other than they believe the Earth to be flat and not spherical as Science has generally demonstrated since before we could send people into space and actually see for ourselves.
The Flat Earth Society's website has some FAQs here for their general thinking on their beliefs which, to me, don't make a convincing case for the Earth being flat. Largely because it includes conspiracy theories that suggest even NASA is being duped into believing the Earth is spherical along with a misunderstanding of how cameras, lenses and the photo-shopping of images work.
That aside, Flat Earther's should consider taking up high altitude ballooning. At its simplest level high altitude ballooning is the process of strapping a video camera to a weather balloon and letting it float as far out into the atmosphere as it'll go before the balloon bursts, sending the camera back to the ground. Maybe try it from different points on each side of the Earth and see how things match up (I don't know... it's just an idea).
I'm not suggesting it as a way to see the curvature of the Earth for themselves using their own cameras, lenses and being able to view photoshop free images, rather, if their theory holds up, it could help make their case.
Though I've also read on their website that it's not up to them to prove that the world is flat. They appear to be of the mind set that others should disprove it. I think it's been disproved without a shadow of a doubt but, I imagine Flat Earther's would likely tell me I drank the Kool Aid.
My only thought is, whatever shape you think the Earth to be, it should be demonstrable with actual facts that either have been discovered or have yet to be. The Earth is here. We're on it. Why not prove it's flat and shut those Spherical Earther's up once and for all?
Has anyone bothered to find that 150 foot tall wall of ice that circumnavigates the entire discworld of Earth in Antarctica? NASA surely can't guard the entire thing when the USA can't even effectively guard the Mexican border?