Transgenderism and Brain Activity

I've been discussing transgenderism recently. One argument that keeps making rounds is that some individuals experience brain activity which is more appropriately identified with the opposite sex than with their own. This allegedly validates said individuals identifying their gender, which is related to but distinct from sex, with the gender usually associated with the opposite sex. A few thoughts:

How is it that one sex has been classified as having certain brain activity and another sex has been classified as having different brain activity if there were supposed counter-examples (transgenders) in the first place? Is the argument that transgenders are outliers rather than examples of the range of diversity within each sexes' respective brain activity? Well, what is the basis for that argument?

Also, the extent to which transgender advocates are willing to argue that transgenderism is more commonplace than assumed is the extent to which this kind of evidence mitigates against their cause. The more commonplace you think transgenderism is, the less reason you have to say brain activity actually varies between the sexes.

If gender is a social construct, when did society construct the idea that brain activity determines gender? I must have missed that. 

Obviously, there is a public policy aspect to all of this. We're already seeing it in whether transgenders are allowed the use of opposite sex bathrooms. The reason we have opposite sex restrooms in the first place is the same reason we don't allow public nudity: it's an issue of public [in]decency.

Do we take people at their word? Or even after people are already applying arising "gender" issues to race, maturity, species, etc., are transgender advocates so naive as to think there won't be abuses - peeping Toms and the like? 

If brain activity really is a sufficient method for determining gender, will professing transgenders be made to verify themselves through brain scans in order for society to have to treat them as such, using opposite sex restrooms and the like? Will gender validation cards be dispensed, and gender policing enforced to ensure no systemic abuse occurs? Apparently not. But why?

I think advocates are afraid that brain scans wouldn't pass muster. Scans would leave out some people who come out as transgenders, and that would lead to accusations of intolerance or bigotry, which is the last thing such advocacy groups want. 

It's very convenient for secularists to be able to shift classifications to suit their public policy agenda at whim. They prefer to be able to broaden the tent than provide real boundaries. Again, as I said above, this eventually undercuts the evidence they will initially use, including brain activity.