All my DC have this, I could write a book about it.
Mind you, we still don’t know where it comes from because neither DH nor I are squeamish or have the same reaction.
So it’s great that you know the source of this phobia, I think that makes it treatable.
DDs are actually ok-ish about periods. Someone else with the same issue told me it’s because the blood isn’t gushing from a wound, I think that’s why the body overreacts with the response of fainting.
She will probably always have this issue and will always have to deal with it, but will have little ways to cope. One of mine mostly just laughs it off now, eg today one of her friends told us she plans on doing nursing, we both just laughed.
I always write a note on their files for school, swimming, dentist, etc. One of mine is exceptionally extreme and will pass out on the spot at the very word “needle” 😑
She has even fainted if another child in school has cut themselves.
By ways to cope, I mean, sitting down / laying down and deep breaths, avoiding certain triggers.
If your dd is very distressed, I would discuss with GP and possibly consider a therapist