"How much of the recent rise in transgenderism can be attributed to people being more accepting vs how much is peer pressure"
Such a study won't see the light of the day any time soon. Yet, I wouldn't classify it as bad faith, as kids/teens are VERY impressionable. (And then there's the whole detrans thing)
Would you classify it as bad faith?
It's not so much that discordant views get silenced, it's that everyone is aware that in the current climate, almost anything can - and WILL - be used to fuel discrimination. Such a study being approved alone would be taken as proof that "even the left finally admits transgenderism is hurting children" by a LOT of people. Even the way you titled the hypothetical study is de facto begging the question.
I mean yeah, these things should be studied. But it's not as though we have no science. E.g. detransition- we know it's uncommon, and mostly due to external circumstances, not regret. In light of this, is proposing to study specifically whether kids are getting pressured into transition(instead of "factors leading to transition"), in disregard of the fact that we already know most benefit from and don't regret it, "bad faith"? Yeah. Kinda.
The science very much so supports the rationale and safety of transition, but everyone and their dog has that one anecdote, that one outlier news story that convincingly prove how dangerous giving people agency over their gender is, that one "but what about the kids!" concern.
Social studies aren't out to find truth. They're out to confirm their beliefs, basically working to justify their own jobs.
[We have the data](https://www.pnas.org/doi/10.1073/pnas.1909367116), being trans is identifiable as early as 2-3yo, from the moment gender differences in behavior become apparent, long before any capacity to comprehend and adhere to complex expectations of boogeyman "transtrender parents".
You want to be treated as arguing in good faith, but fail to research the subject, propose questions that assume the conclusion, bring up vague anecdotes contrary to the statistics you're unwilling to consider, and disregard the fact that transgender children suffer from lack of treatment just as hard as wrongly treated cisgender children.
This isn't what the GP is arguing, did you reply to the wrong thing?
You quoted their question:
> Another one is the question, whether gender dysphoria results from mental illness, or the other way around.
The study you provided - whilst interesting - doesn't answer it or even attempt to answer in its purpose.
The study shows 3-12 year olds who are "socially transitioned and live in families that that affirmed their child’s current gender identity through a social transition" have a strong affinity towards their gender, just as non trans children who have their families affirmation of their gender do.
It seems odd to me you're arguing bad faith when it seems you're acting that way by misdirectly and providing studies irrelevant to the point as evidence of the GPs lack of research.