I’m a doctor. I don’t think you were wrong to take your child to A&E - you were worried, and sleeping all day can indeed be a sign of something worse.
However, as others have said, A&E will have been reassured that her observations and urine dip were normal. They would have been confident that there was nothing that needed treating that day, and it could wait a week. It does sound like that particular doctor did not communicate this information to you well.
Life in the hospital is very tough at the moment. I often cry during my lunch from the sheer horror of seeing patient after patient die. Having to call their families and tell them their dad / husband is dying really takes a huge emotional toll. I am not using this as an excuse for her behaviour, but I think understanding where someone is coming from can make a huge difference.
I think if someone made a complaint about me at the moment, it would tip me over the edge to just quitting medicine. Formal complaints require a lot of paperwork and time on the doctor’s end, not just a quick telling off.
I want to reassure you that I expect the doctor has been spoken to and informed her manner was inappropriate, despite the letter not being overly apologetic. This tends to be how things are, as the hospital can’t apologise in writing without admitting fault and opening themselves up to be sued.
I am sorry it took your child so long to be diagnosed. Coeliac is an often missed but very important diagnosis. The trouble is that a lot of the symptoms are very vague (feeling tired, not growing as fast, looking pale etc). Often these symptoms do turn out to be nothing, and we don’t like subjecting children to blood tests which might make them scared of doctors in the future unless we are really sure they are required. It does sound like this was left too long in your situation though.
In essence, I can totally understand your annoyance, but would gently suggest not following up the hospital complaint, rather sending an informal letter to the GP letting them know what happened. If they are a good GP, they will reflect on this and it may help a similar child in the future.
I do hope your DD is feeling better now!