The "trolley problem" is a philosophical dilemma regarding the morality of allowing harm caused by inaction versus causing harm by direct action. In a common form of the dilemma, you are faced with a situation where a train trolley is headed to hit a number of people. You have the option to pull a lever that will direct the trolley to hit a smaller number of people, thus actively condemning that smaller number.

The argument can be made (though I disagree with it), that you are "more responsible" for deaths caused by pulling the lever than those caused by inaction. Whether or not you agree with this argument, plenty of research has shown that humans are typically very averse to causing known harm by direct action, even though they know that inaction will cause greater harm. That is: "Human instinct" in the trolley problem seems to be to not pull the lever, even though a majority of people agree that it is the morally correct decision.

Finally, the above discussion is more or less completely independent of the number of people on the two tracks. Whether it's 1:2, or 1:1 000 000 doesn't really change the moral argument of action vs. inaction. If you get into the kind of numbers where the number of deaths would have global secondary consequences, you have a new kind of dilemma though. However that's not the case here, hence my choice of pulling two arbitrary numbers out of a hat, rather than using actual numbers for vaccine side effects and infection lethality.

What I'm doing here is noting that the question of vaccinating or not is philosophically equivalent to the trolley problem. Therefore, we can use research of the trolley problem to understand how vaccine sceptics think, and to learn something about how people can be convinced that pulling the lever is the morally correct choice.

It will never be about the science, because we already know they’re completely wrong about everything and therefore they need to make science irrelevant.

I think there's an interesting discussion to be had here regarding cause and effect. You seem to be arguing that these people do not believe in science, therefore they end up as conspiracy theorists, anti-vaxxers, etc. I'm not entirely convinced that's the case. I think it's common that people first establish a gut feeling regarding what is true, then begin disbelieving science when science disagrees with them. Basically, I think the casual chain is inverted relative to you. Of course, once people have started disbelieving science, the rabbit hole is open.

the supposed strat of… agreeing them out of their position(?)

That's not really what I'm arguing for. My point is that it never strengthens your position to disagree with things you know are correct, just to try to make a point. If someone says "water is denser than oil, therefore 1 L of water is heavier than 2 L of oil.", it doesn't help you to start by disagreeing with the initial statement. Likewise, when someone says "vaccines cause dangerous side effects, therefore ...", it doesn't help your position to disagree with the initial statement, which is well documented to be true. An immediate example that comes to mind is that the Astra Zeneca (I think) COVID vaccine caused several deaths due to triggering a very rare heart complication (if I remember the details correctly). If your initial counter argument is "vaccines don't cause deadly side effects", you can immediately be brushed off as having no idea what you're talking about, since it's well documented that they do. Any successful counter argument needs to start by acknowledging what is correct in the initial statement, before dialling in on why the conclusion is wrong.

The point of the trolley problem is not to use it to convince the vaccine sceptics, but to use research on it to understand how they think, and thereby understand how they can be convinced. Surveys have consistently shown that not all people agree on what to do in the trolley problem situation (around 10 % choose to do nothing). It's only a false dilemma if you view it from a utilitarian standpoint, but the whole point of the thought experiment is exactly to compare a utilitarian standpoint to other views. There's quite a bit of research regarding the psychology of how we experience the consequences of action vs. inaction, and it appears that humans instinctively prefer inaction over actions with negative consequences, even when inaction carries more negative consequences. The book "Thinking fast and slow" by Daniel Kahneman goes into some detail on this. I think a lot of vaccine scepticism is grounded in this instinctive response, which in turn causes people to disbelieve science due to confirmation bias.