Uh-huh. Body dysphoria is indeed a REAL disorder (or, more accurately, it's the product of a real disorder). I don't know quite whether you think anyone actually disputes this.
But it has now been determined by several globally-important organisations, and by many major governments - based upon scientific data and expert opinion - that gender dysphoria is a genuine medical condition in itself. Not a disorder, nor the product of a disorder.
Gender dysphoria is listed as a diagnosis in the DSM5 which is the Diagnostic and Statistical Manual of Mental Disorders. Therefore it is classified as a mental disorder as are all diagnoses in DSM regardless of whether the word 'disorder' appears in the name. It was redefined from gender identity disorder to gender dysphoria when DSM5 came out in 2013, in order to remove stigma attached to the term and to describe the condition more accurately. Some trans activists did not want any diagnosis related to transgender identity retained, but the consensus was to rename rather than remove it. If it was regarded as a 'medical condition' rather than a mental disorder (where does this claim come from?) it would have been removed.
WHO followed the change in DSM terminology which is usual practice. They agreed to place it under sexual health rather than mental disorders, which is unusual for DSM diagnoses and reflects trans activist lobbying.
It is probably not appropriate to describe gender dysphoria as a mental illness or the product of a mental illness. Many other diagnosis in DSM are not appropriately described as mental illness either (e.g. autism). Evidence of a neurological basis for a disorder does not automatically affect it's inclusion in DSM.
(And, incidentally, mainstream medical thinking to this effect is precisely why gender reassignment surgeons are not required to consider the "do no harm" maxim - in the way that, for example, surgeons would be required to do if someone with body dysphoria wanted their leg to be amputated. The (e.g.) removal of the male genitals of a trans woman who has been properly assessed and diagnosed (provided, of course, that this is what the trans woman wants) is the treatment of a valid medical condition, as opposed to being the treatment of an illusory belief that is itself the product of a mental disorder)
But both gender dysphoria and body dysmorphia are DSM diagnoses as above. Allowing sex reassignment surgery but not the amputation of a leg in this scenario could be seen as inconsistent, since the purpose of both is to alleviate distress. However, I suspect that the sex reassignment is viewed differently because it does not involve making somebody disabled in the sense that a missing limb is a disability but having a particular sexual anatomy is not.
Even if you were correct in the significance you seem to be attaching to changing the name of a DSM diagnoses, it cannot be related to allowing sex reassignment surgery because the surgery has been allowed long before this change was made, when the term gender identity disorder was still used.