TLC - Unpretty Lyrics Meaning

I think this song is heavily inspired by a politician's wife or a female from a politician's family.

You can buy your hair if it won't grow,

Hair is often used to symbolized roots particularly heritage, here buying hair is most likely overemphasizing or exaggerating something that is minor or not well defined. Like a Spanish great grandma who you never knew while she was alive but are "supposed" to be similar in spirit and manner.

Then there is the fixed your nose if he says so. Nose often represents face, appearances, or perceived image, on the nose; flaunting an image or behavior that is usually stereotypical, cut your nose to spite your face; denying a heritage or background for a cause or using a heritage or background to cause damages to the heritage or background itself.
Sounds like he is telling her to act or behave a certain way in public.

Every time I think I'm through
It's because of you
I've tried different ways

This is actually super common in PR and Image Consultant. One doesn't just prop up a fully developed image in public perception, one unveils in stages something uncomfortable maybe discarded for something similar as long as one image is overall consistent. Sounds like a newly married young wife of an ambitious politician or a young lady from a political family being groomed.

Finally this is sort of far fetched for me but I will leave it up to you to decide.

My outsides look cool
My insides are blue

While both blue and cool are words used to describe coldness, here it could mean polite but detached behavior. Which sounds like isn't her. Also something else that inside? Blood. Blue blooded is nobility and in modern day often used to pointedly at the higher income families and especially politicians.

Notable mentions,
I was told I was beautiful
But what does that mean to you?

Often before a press conference or something important the subjects would be fluffed etc saying something to boost their own confidence. "You are a confident, good looking, successful man." For females it's often "you are beautiful" but due to how commonly they are called "beautiful" it's often been cheapen to not be a confidence booster.
Very often these young females feel being beautiful is the norm and standard amongst the elite and they are just average among their peers.

Something else,
Look into the mirror who's inside there
This in particular sounds like a mask that sometimes fits so well she often forgets she wearing it, other times it feels like a heavy burden upon her face, this makes her confused between the woman she is perceived to be, the woman she wants to be, and the woman she is.

Great song though.