To be fair, hair dye does have a decent history. And if a person is only allowed to change their hair color, then you can say, "No, I still recognize your character with red hair. If you want a more effective disguise, I can help your character lop off their nose.
"
Of course, there would be people who use the hair dye frequently, but that's part of the point. I think it'd fit right in at port, it'd work as another gold dump for characters, and it'd also would give people another thing to rp with.
You mention the "you can't recognize my character!" thing as a negative, but I can see some creative uses for it. A person could purposely dye their hair black, commit a robbery or a mugging with a mask, but leave their hair exposed, and then scurry off to dementileu to get their hair back to their natural color. (either as a frame-job or just to help with their disguise)
It's the "bleach my hair in the gas station sink" of the fantasy world.