If it is an important part of your story then I would suggest allowing CC for skin tones in the similar hue. Like if the character is fair, then allow the user to CC the skin tones from shades 00-04 or if they are of darker skin tones then allow for CC in shades 05-10 in all the shades.(neutral,rose,gold,copper).
I personally dislike limited CC as it feels crippling to not be able to choose your own skin colour and play it as other skin tones, though when the authors provide variation in the shades then it’s nice to play because then we can customize the character to our liking.
You can also allow full CC then add a warning about changing the skin colour and that it has a big role in the plot so it’s advisable to not change the skin colour. Then the user will probably not change the colour and everything will be fine and even if they do then it’s their problem.