Suppose in some cases gender dysphoria has a neurological basis. Would that justify making special accommodations for the transgendered? No.
i) To begin with, due to neuoplasticity, you can't always tell which comes first. Although the brain affects behavior, behavior can affect the brain. To take a comparison:
Does playing the violin affect the brain? Most violinists would probably say “yes” in a subjective way, but there is striking objective evidence, too. Perhaps most interesting are the findings that the brain is actually modified physically by studying the violin in ways that make it easier to learn more. The cerebral cortex, the site of higher thinking in the brain, is not a static structure. Its organization changes over time, giving the brain an astonishing ability to adapt to new needs.
Broca’s area is a small part of the cortex which handles many tasks of spoken language and musical abilities. The amount of gray matter (neurons) in Broca’s area is larger in musicians than in nonmusicians. In fact, the volume of gray matter in this brain region increases as the number of years of playing increases. In most people, the amount of gray matter in Broca’s area decreases with age, but in musicians, this does not happen.
Are they violinists because they have musical brains, or musical brains because they are violinists? Apparently the latter.
So even if some people with gender dysphoria had abnormal brains, that, in itself, wouldn't tell you if that's the cause or the effect of transgenderism. Does it originate in the brain? Or does the mind and attendant behavior condition the brain in that direction?
ii) Assuming gender dysphoria is real, it's not like accommodating people in wheelchairs. For generations, people with transgender feelings have been using restrooms and locker rooms that match their biological sex. They are perfectly able to do that. Indeed, they've been doing that all along. It hardly justifies unisex restrooms, locker rooms, sports teams, speech codes, &c.
iii) To take a comparison, suppose some phobias may have a neurological basis–which may well be the case. That doesn't mean we should upend society to create special accommodations for people who suffer from phobias. They are entitled to our understanding. But social existence requires a large measure of stability.
To take another comparison: most people are right-handed. As a result, many things are designed with a right-handed bias. I'm sure that's annoying for south paws (my father was a south paw), but if you can't always design things neutrally, it's sensible to make the majority the default setting.
BTW, that's an opportunity for enterprising businessmen to take advantage of that neglected market niche by designing things for south paws. Let the private sector solve the problem.
iii) Assuming gender dysphoria exists, and assuming it has a neurological basis, it would be a neurological disorder. The question is how to deal with people who suffer from neurological disorders. It depends. People with cognitive impairment may require extra assistance, supervision, or protection.
However, someone like Bruce Jenner, to take a prominent example, is clearly capable of functioning quite successfully in normal society when he wants to.