My name is Pauline and I am 37 and I have EHK. I don't know what I can advise you in regard to other sun screens, because to be honest I have never really used them, and when I did try one it wouldn't rub into my skin, it just sat on top. I also live in Australia so our products may not be available in the US. I keep out of very hot sun, but can sit in mild weather without a problem.
I guess your daughter having red hair makes her skin more prone to burning, so she should probably do activities in as much shade as possible. Overheating is a major problem with the sun, and I think we tend to automatically escape the sun when our body tells us we have had enough anyway. Ensure she has plenty of fluids, especially when it is hot.
Is or would your daughter be happy to wear nice lightweight long dresses with long sleeves in the summer, something that she would be comfortable wearing in her age group. I suppose some kind of compromise would be ideal where she could wear shorts and singlets if she wishes on milder days, and keeps the long sleeved dresses for very hot days.
I am sorry if I have not been much help to you, but these are just my thoughts from a practical side. Have a look through the posts and I am sure you will find some articles on overheating and sunscreens.