My daughter is a similar age and I would also describe her freckles as similar to your daughter's. I didn't know if the derm would respond the way your's did when I took her, but she turned out to be incredibly understanding and not at all dismissive. I think my GP took particular care to refer us to someone who she knew was sympathetic. I know people have lots of opinions on hydroquinone, but at the end of the day you can buy it in lower concentrations over the counter (in Australia at least), and apply it as often as you like without any medical supervision at all. And in some Asian countries lightening creams are very popular and the shelves are littered with hundreds of skincare products containing hydroquinone. So to me, having it prescribed in a higher (but still low) concentration, and applying it sparingly once or twice a week under medical supervision sounds like a reasonable option when the freckles are affecting her self-esteem - even if she is a child. Good luck with your daughter. I can only suggest further investigation to see if there is a more sympathetic dermatologist out there for her.