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In order to get support for a compassionate reassignment, the specialist needs to state how moving you is the only possible option, not the best option, or the preferred option.
Also, your command will need to specify how retaining you is in the best interest of the Army, versus separating you for family hardship.
The issue here is that your spouse made a cry for help in order to bring you home from Korea. What would prevent her from doing the same every time you deploy?
Second point is that compassionate reassignments are a temporary solution for a problem. This goes back to why your command needs to explain why it's best to retain you and not separate you for hardship.

SPC (Join to see) your profile doesn't have your updated assignment information. How long have you been in Korea? 10 months 2 days used to be considered a completed overseas tour. At that point HRC and USFK are happy. As a Company Commander I tried if it were at all feasible to get soldiers with different difficulties to the 10 months 2 days so the overseas short tour block was checked, OSR, KDSM, and they could move on. That would keep you off the short tour radar for a few years.

Did you accept a COT? You're profile shows Graf in Germany since 2014.

The essence of a compassionate reassignment is how your presence will improve the situation, and how dire the situation is. The trick is that you have to walk the line with "being there" and being deployable and able to handle your soldiering obligations. So there should be some gist of how this is temporary. I think the minor child is an important factor in favor of a compassionate reassignment.

I was looking for a thread that had an NCO that worked compassionate reassignments at HRC, she was a wealth of knowledge. Of course I can't find it.

Get your documents and most importantly, the letters from her attending providers. Is the issue simply reunification or does it matter specifically where you are assigned? If you both get to say, Ft Campbell, does that alleviate the issue?

Submit the paperwork as requested by HRC. My experience with these was that they were rarely approved because the burden of having to justify that the Soldiers return was the only remedy for the situation.