She didn't lose the deal over a photo... that's just a ploy to get you to cave in. I'd correct it, send it to my client, and quit discussing ANYTHING with the non-client party! It takes back bone to be in this biz... and you just admitted to more liability than a $200 fee.
You don't send HER anything! You shouldn't be discussing this with her. Send the corrected report immediately to your client, not the borrower. Since the picture had nothing to do with the loan, and it was a clerical error only, there's no need for a refund either.
Wade,
Why did you not collect the other half of the fee? you said
"Borrower's loan didn't close and she paid half of the appraisal fee upfront. She wants me to refund her the $200 because it was "my fault" that the loan didnt close."
Like others have suggested, correct the photo, send to your CLIENT with the invoice for the remainder of your bill.