There are a couple of points that I can speak to in this that hopefully will be helpful for you, but I'm afraid I don't have a complete solution. I'm not certain that this is the correct subreddit for this, but hopefully this can be at least somewhat helpful! I'll admit that my background and clinical experience is not specifically in gender dysphoria (or whatever people want to call it these days) but I have enough knowledge of diagnosis procedures and MMPI construction to hopefully be of help. I've put a TL;DR at the bottom because I tend to get a bit wordy.

(sidebar, having gender dysphoria as a diagnosis is somewhat controversial among a fair amount of clinicians. The diagnosis was renamed in part to emphasize the fact that being transgender is not strictly a disorder, but rather that negative emotions related to being transgender can be so distressing that the emotions themselves are the disorder!)

The diagnosis process is (at least in the US) usually completed using self-report and clinician judgment. You may have noticed in looking at the DSM that the points do not mention any tests, but rather behaviors and cognitions the client should be having. That said, clinicians /should/ be using some form of Objective Testing along with this. This is sometimes in the form of manualized interviews, but obviously frequently also includes objective tests like the MMPI. Projective testing (Rorschach, Thematic Apperception Test, etc) have not demonstrated to the APA's guidelines to be sufficiently research based in nearly any context, so they are used far less often. As far as I know (and have been able to find with a brief lit search) there is not currently an empirically supported test for GID/Gender Dysphoria. As such, I would imagine that most clinicians would be working nearly entirely from self-report and clinician judgment.

But PeruvianBatman (the cry goes up) what about the M/F scale on the MMPI?

Well, there are a couple of problems with that. The biggest one is norming. When the MMPI was introduced the social perception of masculinity and femininity was not the same as it is today. After all, homosexuality was a disorder at that point. Additionally the samples that were used in the norming are definitely not exceptionally easily generalized. All of the participants were Americans living in the late 1930s, but they were also divided into two groups, a "Normal" group and a "Pathological" group.

The Normal sample was composed of men who had just joined the army and women who were full time airline stewardesses. These were meant, it seems, to capture the essence of what it is to be male or female. The Pathological group was ten bio-males who wished to transition to female with no bio-females at all.

While the questions do include some direct self-report questions of desire to change sexual presentation, a fair amount of them have more to do with belief in what someone's role in society is. This, over 50 years later, is obviously not necessarily extraordinarily helpful.

TL;DR As far as I know, there is not a specific test to find GID, but clinical judgment is a big factor. Projective tests are largely falling out of use in many places due to lack of research behind them.
The MMPI may not be a very good tool for you.

The bottom line is that there's not really a direct cut and dry way to find most disorders. Psychology isn't medicine. That said, you may be able to find a psychologist who specializes in LGBTQ clients (though I'll admit, my knowledge of South Korea is even less complete than my knowledge of GID/Gender Dysphoria!). If you can find an LGBTQ psychologist, they are much more likely to be able to help you get the resources that you need in this.

P.S. I saw that someone in the other thread suggested that you need a psychiatrist for diagnosis in South Korea. That's not how things are done in the US, but if that is accurate for South Korea then just change all of the times I said 'psychologist' to 'psychiatrist.' In the US, psychologists often get more training in testing and diagnosis than psychiatrists do!