Hi HappyToBe - stressful situation for you I can see - and I think it's always harder worrying about someone else's health than your own.
I can tell you from personal experience though that it really does take a long while to get to the stage of being an emergency due to over-heavy periods - and by "over heavy" I mean things like going through a super+ tampon AND a max absorbency pad inside an hour (inside 20 minutes at times!) and regularly being in that state for 5 days at a time, and on several occasions having bleeds that lasted up to a month - my first overly heavy period was in July 2014 - and things got steadily worse for several years with symptoms getting worse and worse until I did indeed end up in the emergency situation you fear last October - but that took three years of VERY extreme problems to get to that stage. Absolutely not wanting to trivialise your daughter's situation but I did think that maybe knowing how long it takes to get to that stage might help you worry a little less as it really is highly unusual for things to get to that level. I was in a very severe situation but I'm still here to tell the tail!
Why not suggest to your daughter that she considers taking a daily "WellWoman" or own brand equivalent tablet - the Boots own brand include 100% of the daily requirement of iron which will help, it might also help her to keep a diary of her periods so she really gets a feel for what is normal for her - she could do this in a "notes" app on her smartphone perhaps - most of those can have password protection used to keep them from prying eyes.
If you can get her to go to the Dr's then go with her, and ask about the contraceptive patch which is a good option for someone who'd not great with taking tablets or anything more invasive - I used it for years as a result of a poor memory meaning that taking the pill wasn't a good choice for me, the surgery didn't much like it but I held my ground on it and they did prescribe. It does make periods lighter - MUCH lighter in fact, and regular as clockwork which can also be very helpful to reduce anxiety around them.
If you're not comfortable talking to the GP about anxiety medication for yourself then maybe try something like Rescue Remedy as a starting point? A lot of folk swear by that. I have to say though from what you've said it does sound as though a chat with the GP might be helpful. You might also look at a Scottish NHS website called "Moodjuice" - very helpful indeed on the subject of anxiety.