She's 17 and so it's her decision.
I was diagnosed this year but I am -a lot- older. I never got given the numbers for my blood results, but I did have the endoscopy. I didn't have sedation on the recommendation of a (female) colleague at work who has had several. It was pretty unpleasant but it only lasted 10 minutes and no needles were involved. There was a numbing throat spray but the main problem for me was that I struggled with supressing my gag reflex -- apart from that it wasn't really painful. They didn't even make me change into a hospital gown. I've had worse times at the dentist.
If it were one of my dds then I would stay with her during the procedure and hold her hand and mop her brow. I'd also check the method used for sedation - my impression was that if I had opted for sedation it would have been tablets rather than an injection, but as I didn't take it I am not certain.
The endoscopy does a visual check and a biopsy. I was told it's unusual to see coeliac on the visual check, which is more to make sure that there aren't other things going on, like ulcers or cancer. The biopsy samples the tissue for microscopic examination to look for changes in the villi, which are like very tiny fingers poking out from the surface of the gut. In coeliac disease the villi are shortened or eliminated in the autoimmune reaction to gluten stimulation. Cutting out gluten allows the villi to heal and restore, but this apparently takes some time.
I am only a patient, not a doctor, but villi don't heal overnight and I would doubt that all signs of damage would disappear in 5 to 6 weeks. This level of symptoms is very bad. I am pretty sure with hindsight that I have been coeliac all my life, and have the dental and nerve damage that comes with it, yet I never approached that level. You may be able to get away with putting her on a GF diet immediately without prejudicing the outcome of a biopsy.
As I have commented on here before, I think it is better to have a diagnosis on your medical records. That means you will, as you say, be eligible for the special treatments and it means that you can legitimately demand GF in workplaces and if you are a hospital in-patient. Ultimately, though, there are a lot of stories on here and elsewhere from people who simply went GF and their symptoms went away. Foods on prescription are being taken away -- my local CCG refuses to allow them (in breach of national guidelines) and the national guidelines are becoming more restricted with time.
Fundamentally, she has to decide whether she wishes to pursue this to diagnosis. If so, and given her symptoms, there are two avenues to think about. One is to go back to the GP and ask the GP to ask the consultant for this to be escalated to urgent. The other is to go private - a private diagnosis will be just as valid and a private doctor may be more sympathetic to her age and associated difficulties and be prepared to diagnose on bloods alone.
I came at it from an unusual angle. I was being investigated in relation to the nerve damage and it was the neurologist who did the wide-ranging blood tests to find out what had caused it that threw up coeliac. The GP referred me to the local gastroenterologists as soon as the neurologist's report cam through, but then I hit a bit of a brick wall. Because gastros don't actually treat coeliac, the department clearly saw it as a bit of a nuisance and pushed me (and others) way down the list. That's why, in a case such as your dd's, if I was able to afford to go private I would seriously consider it.
Sorry for rambling but I hope this helps.