6500K is correct.
Tell me about your room lighting.
Eizo's "printing" setting is literally for printing. That is, the commercial offset printing industry. NOT for photography.
Which lab?
5000K is not correct. If you have a screen-to-print match at 5000K you are either using a dreadful lab, or have dreadful lighting in your room.
The best I can suggest is to add a Solid Color layer of hair brown, and set it to Multiply mode, then mask it on very gently where needed.
I used a 10% brush in this example:
I realise it's not spectacular, but I hope it's better than nothing.
If it's going to be possible (and I'm deeply skeptical) there can't be those black shadows. Start again with your raw processing, without ANY black clipping.