Of course you can insist she goes home on a Friday! If she is at serious risk of a reaction, then you can certainly demand that she is not exposed to this risk. Does she have an allergy specialist who could help by talking or writing to the school?
However, she may be able to stay in school. It is perhaps worth checking this out. Sitting with her (near the door) on a Friday lunchtime may not be a bad idea. If she does start to react at all, then the school are likely to take you seriously once they have seen this - but she may be okay in a big lunch hall which is well ventilated.
Ds1 has multiple food allergies, and we are lucky that his school take this very seriously. He is in reception. I had to bring him home for lunch until Christmas last year, while the teachers and support staff trained all the other children to have their hands wiped after lunch, and to wear cover-all aprons so that they didn't get smeared with yogurt and banana etc. Now that ds is allowed to stay in school for lunch, he has a support worker who sits next to him during lunch times.
One of his good friends in reception has a severe fish allergy. The friend's mum is Bangladeshi, and fish would be a main part of their family diet except that her little ds reacts to even the smell of her cooking fish in their home. This boy has never had a reaction in school, even on fish Friday, perhaps because the dinner hall is so big. However, she has never pushed for proper diagnosis and support, so he does't get the same level of support as my ds.
I wish you much luck with this. Don't be afraid to keep nagging the school, and I do think that calling the anaphylaxis campaign is s very good idea.