I think that the Flat Earthers can be separated into two types:
1. Those who wholeheartedly believe in it, have made up hypotheses for it, and generally suffer from too much confirmation bias to convince. Science is extremely complex, and many of these people do not understand that even a PhD in physics only understands an extremely small fraction of current science. Many of their clever rebuttals are trivially shown (and have been shown long ago) to be fallacious. Their knee-jerk reaction is to attempt to disprove modern science without a full understanding of it. Take, for example, Tom Bishop's assertion that if you hold your hand so it's at a higher angle ("above" from your perspective) than a distant lamppost, the lamppost looks upward to see your hand... that's false, and obviously so. Or his various machinations about Doppler shifting from the stars (like randomly looking up the term bathychromic shift without understanding that these apply to molecular spectra). That's not to say Tom's a bad person. He's actually one of the better people here, because he engages constructively in debate, and a lot of wrong stuff he says not out of bad faith, but of ignorance. He's just heavily affected by these psychological biases, specifically the backfire effect and confirmation bias. Or you can take a look at Pete Svarrior's nitpicking at arguments while denying the obvious truth through circumlocution and technicalities. That's called muddying the waters. These are the people who are heavily affected by the psychological biases you talk about; they perform mental gymnastics because in their minds, they cannot possibly be wrong. But you should remember that they could say the same about us, because from their perspective, they think they know the "true science" and that we're being dogmatic. Of course, this is not true (precisely because of all of the scientific verifications that they're ignorant of as a result of almost no physics education), but it's an understandable thought. These are the people who, short of throwing them out of an airlock with a space suit of their own making, and then interviewing them afterward, will never believe in a round Earth. This is because they've gone so deep that their minds have closed to learning science (they question before they understand what they're actually questioning), and without a reasonable degree of scientific knowledge, one cannot ascertain the validity of FE or RE as currently presented.
2. These are the people sitting on the fence, usually because they don't understand the entire debate and from a layman's point of view, a lot of what FE says makes sense. While those who usually respond to the various threads in FE debate are mostly part of the first class, I spend time rebutting them not to convince the people who are actually debating. It's not untrue that in debates, both sides usually think they won. But for someone sitting on the fence, reading the various debate threads may at least convince them to second-guess the fallacious explanations of those without expertise in the relevant parts of science, and take a skeptical eye toward hypotheses/explanations that are widely panned by scientists worldwide. The main point here is that science requires domain-specific expert knowledge to critique; for example, you cannot critique Special Relativity without understanding all of the experiments that went into making this rather unpleasant theory. Of course, the flatness of the Earth is a rather irrelevant debate to today's society, as Flat Earth believers will never work on space exploration, long-range navigation systems, and space-based telecommunications by virtue of their complete wrongness on several levels of science. But hopefully people on the fence of Flat Earth, and then convinced of the rigors of scientific study (even if it's impossible for them to understand), can apply this elsewhere. Maybe the next time they see a conspiracy claim (vaccines cause autism, chemtrails, global warming hoax) that's not as clear-cut (I mean they're pretty clear-cut, but nothing compared to the flatness of the Earth), they'll think twice about "doing your own research" (from poorly-made YouTube videos chock full of technobabble or fallacious reasoning from obvious non-experts) and then choose to actually inquire through reputable sources (and rely on them for scientific reasoning, while still leaving the standard logical deduction to themselves.
TL;DR: you shouldn't heed any of the meta- arguments of Tom Bishop, Pete Svarrior, etc. They're irredeemably mistaken on their belief that science is simply a fallacious appeal to authority. This is because they aren't aware of and don't understand all of the evidence that was collected to back up current scientific understanding. This is a result of a lack of a true understanding of the basics of science.