[0] A sex difference in the human brain and its relation to transsexuality http://faculty.bennington.edu/~sherman/sex/TRANSGENDER.pdf
[1] Male-to-Female Transsexuals Have Female Neuron Numbers in a Limbic Nucleus http://press.endocrine.org/doi/full/10.1210/jcem.85.5.6564
[2] Regional gray matter variation in male-to-female transsexualism https://www.ncbi.nlm.nih.gov/pubmed/19341803
In any case, it's actually more-or-less irrelevant in arguing whether a trans person is actually the gender they identify as, and is mostly a curiosity until it leads to more focused treatment. The bit that does matter is the last question - am I who I am in my head, or am I who you think I am?
Source?
If the difference is so clear in an MRI why not run a simple blinded study where researchers look at brain scans and tell you who is transgender and who is not? Such a study would not only prove this theory outright, but would likely make a researcher's career.
No. You can think yourself into a different gender. Gender isn't sex - this has been written about since 1963, and is generally agreed on at this point - gender is a combination of personal and cultural mental attributes, while sex is generally "what your body physically is". This messes up slightly when you get to certain sex-driven gender attributes, but, in general, it's a reasonably clear line.
Sex is mostly irrelevant unless you're intending to sleep with someone or you're a doctor, and even then what you're asking about is generally a certain set of characteristics where a number of people fall between the lines. If you're a trans woman who's undergone GRS and you say "my sex is male", that's going to confuse an awful lot of people.
Edit: My specific point, in this response, is that identifying as a gender which does not match your sex is not "playing pretend" or less important somehow, as you insinuated in your original post.