Tasa, when I was your duaghter's age I began to hate my hairy arms too, though have to admit it wasn't till was 10 or so that I got really self conscious. Children tend to grow up quicker now, don't they?
I was very pale skinned and had unusually dark hairs on my arms. I didn't come to terms with them till I was in my late teens, and spent years bleaching them, shaving them, but mostly covering them up. I was a determined cardigan-wearer just like your dd. Even when I did bleach or shave I was still very self conscious. My arms now had lots of bleached hair or stubble or nicks from shaving. So still not like the other children's arms.
I don't know what I'd advise on the bleach or shaving front. There are far more products around now. If you go for shaving as I did from time to time, I can reassure you that my arm hair never grew back thicker afterwards. And as I've got older and had children my arms seem less hairy - hormal changes perhaps? or do I not care as much?
Anyway when I was younger no amount of reassurance from adults helped me get over my hairy arms, though it was nice to hear them trying. I wasn't stupid - I could see I was the only girl in the class to have so many hairs on my arms. It was a fact. I couldn't be talked out of it.
I always chose clothes to hide my arms. It was really important I had some cover. I could then choose to reveal my arms if I wanted to. The best thing my mother did for me was to buy me the cardigans and long sleeved things that I wanted, and then let me wear them when I wanted to, without commenting on it, even if the temperature was in the 80s.
So IME, let your dd wear her cardigan for PE if the school allows it. Don't make a big thing of it. When you are both shopping for clothes, let her buy long sleeved things - even take her on a special shopping trip to do this. With all the different fabrics and fashions around, she's bound to find some nice hot weather stuff. Don't mention her arms, just let her get what she wants.
HTH