Nah don't allow yourself to get defensive.
Ask yourself is it reasonable that she wants to see a picture of you without a hat.
If you think it is reasonable..........(it is) shoot her a picture without a hat.
Maybe she fears you are a ginger?
On a side note, I allowed the mentality of this forum or red pill to influence a response I gave to an attractive female:
We exchanged whatsapps and she sent me a photo (there was no need I know what she looks like) and said to me "It's because I have no profile picture (on whatsapp) and you don't either" So I just shot her a picture. I got a "wow" from her (in a pic that I think I look tired as hell!)
Then a while later she asked "what height are you exactly?" It made me think of the posts on height here......I'm 5'10 and I could give a fvck........normally I would have just said "5' 10" and that's it
But I didn't, instead I said "why, what weight are you?" of course she was surprised and said "that's not a question that you ask a woman!"
I continued the red pill manner and copied her question and asked "well, is this a question you can ask a man?" (now, I probably shouldn't have, because it really doesn't bother me at all"
What did the poor baby do? She sent me her weight and her height and apologized if you caused me offense.........she said "ah it was just the photo I thought you were really tall in that photo and was curious"
So put me in a little bind as I wasn't offended at all but my response gave the impression that i was and kind of killed the vibe...
Avoid being defensive I say. She either likes you the way you are or she doesn't. Can't please them all same as they can't all please you.