We can debate the classification until the cows come home. Personally, I don't think it matters to most people outside of researchers and, as others said, the fight to say it isn't mental seems to be more about the stigma of mental illness than how to classify those who are distressed by their sex characteristics and wish to be seen as the other sex. I've read that there is more evidence that PTSD alters the brain and hormones than being trans (though there is still overlap with the general population so still no way to tell just by assessing either) but few would argue it isn't a mental illness.
The question I think should focus on who should be medically treated and how and best practice with our current knowledge starts with common mental health treatments and assessments before moving to anything physical like surgeries or hormones. Make of that what you will. Beyond that, it shouldn't matter what causes it when it comes to the bigger social questions around it.
However, if we want to debate it, there is no evidence that a trans person, prior to hormonal treatment, has significantly different hormones than anyone else of their sex and pretty much all brain studies are done with those post-hormonal treatment which affects the entire body including the brain as they're powerful medications. There is no way, looking at brains or hormones, to tell who is or is not trans pre-treatment. Hormones play an important role in the entire body as do all of our chromosomes, including our sex ones, which alter how many hormones - not just our sex ones - are processed in many parts of the body. That's why many diseases show up significantly more in one sex than the other even outside of our reproductive organs. There are many biological differences between the sexes - two females will always be genetically close than a female compared to a male, regardless of any other factors (though some men have significantly more tits than I've ever had). I have not seen evidence that that has anything to do with being trans though. And really, mental and physical aren't entirely separate, many conditions have intertwining both - our mind is not entirely divorced from our bodies, our bodies are the only way we exist or experience anything.
Also, yes, to those who think it's clever, please don't bring intersex people into trans discussions like debate pawns for trans advocacy. There are male intersex people and female intersex people. Intersex people are those who have a disorder of sexual development, our reproductive organs and other parts of our endocrine system grow wrong - some to the point of being lethal without treatment. Some intersex conditions kill people - that's basic information about it. Does that sound like a normal part of a distribution spectrum of sexes or a medical concern? There is no intersex gamete, having no gametes has never excluded someone from their sex, there is no intersex sex or sexes, we do not in any way disprove human dimorphism (in fact with the range of medical issues unrelated to reproduction intersex people are at high rate of having, we only seem to prove how important it is to health) and - as been studied over and over - we are no more likely to be trans than anyone else, there is no connection between gender variances and being intersex and people really should stop acting like intersex conditions prove anything about trans people. It doesn't, they're unrelated, and it's very tiring being either a gotcha or pitted against trans people.