Is being transgender a mental illness?

My friend was saying how transgender is no different from Body Integrity Identity Disorder (BIID).

From Wikipedia, “…is a rare mental disorder characterized by a desire to have a sensory or physical disability or feeling discomfort with being able-bodied, beginning in early adolescence and resulting in harmful consequences.”

Basically, people with BIID want a part of their body like an arm to be amputated. He said its obvious BIID is a mental illness and I agree on that. Then he explained that the treatments/solutions are either changing how you think through methods like Cognitive Behavior Therapy or amputation. Amputation has a 70% success rate and is the most effective current solution.

I think the best solution for transgender is also surgery and hormone treatment. He said that just like amputating an arm removes functionality, amputating your genitals removes functionality. He said just like if a person had their arm amputated for BIID they still have a mental illness, and that its no different from transgender people going through surgery.

I had no counterargument to this and I think he is logically correct, am I missing something here?