There is actually much well-conducted scientific research in this area, so I may have misunderstood your point. What exactly cannot exist, and why not?
You can't do a controlled experiment; all you can do is observe while myriad arbitrary changes in society take place. So, an apparent increase in gender dysphoria *may* be due to 'mass hysteria' caused by a single episode of a soap opera. You'll never be able to disentangle the factors.
But I'm sure that there are "well established scientific methods etc.", and it's possible to write lots of words that look very sensible and scientific, and to list lots of statistics using methods that are 'by the soft sciences book' thus creating a piece of 'well-conducted scientific research'. But something that is conducted as well as it is possible to do doesn't make it valid. And as scientists always point out at stage 2 of the argument, science doesn't purport to reveal the truth; it can always be superseded. And so the circle completes.
Can you think of an example, either historical or hypothetical, where a claim of sexism would be acceptable according to your standard?
A claim can always be made, but the response to it in this area must always be subjective.
You can't make a case for gender being a tremendously important difference between people and then say that men and women won't spontaneously organize themselves on gender lines. If they do that, the statistics will show gender 'imbalances' and any attempt, no matter how correct by the (soft) scientific book, to disentangle genuine gender differences from sexist behaviour will have a strong subjective element to it. The fact that 'toxic masculinity' is now a common term amongst 'progressives' doesn't bode well for my gender's (or my sex's) chances in the future! I don't think scientists and politicans are showing themselves to be unbiased when there's virtue capital on offer.
How do you distinguish between "genuine" mislabelling and "misattribution"? What is the decisive difference in your view?
I don't think I can do it.
In the case of an unfortunate person with dysphoria about a body part and who wants to chop their own leg off, it is fairly easy to see that this is a mental aberration and they should probably not be 'indulged' without extreme reluctance - and I don't think we would hold out too much hope for a complete 'cure' afterwards.
Can you distinguish between a person who attributes their depression or unhappiness to their gender label rather than a body part? Might it not be more-or-less the same phenomenon but with just a different fixation? No doubt you can say common sense things like "I would want to talk very carefully to the person at length in order to ascertain their genuine identity issues before recommending any further actions. This would be done in accordance with well-established research into the field". Etc. But this would just be words. The person who wants to chop their leg off would be just as sure; determined; lucid as the gender dysphoric person.
Luckily, the person with the fixation on their label *can* just change the label - or will be able to in the near future. If they are happy forever from then on, then I am glad.