Right Ariane, what can I do to help your stress levels.....? Hope this helps.
I have done a lot of research about neurally mediated hypotension www.blood-pressure-updates.com/bp/low-bp/low-blood-pressure-treatment/neurally-mediated-hypotension.htm and although I am not doctor I would stake all my chips on your dd1 having either this or postural orthostatic tachycardia which I have linked below. It is directly associated with EDS hypermobile type, you have described exactly what I do, especially getting up too quickly, being in too hot an environment and being much more likely to suffer with it if I am unwell, stressed feeling depressed etc. There is a new eds website I found it might help. www.ehlersdanlosnetwork.org/hypermobility.html
It also talks about postural orthostatic tachycardia, en.wikipedia.org/wiki/Postural_orthostatic_tachycardia_syndrome Most importantly the heart is completely structurally normal. I spoke to my ds's rheumatologist about this last week, and they admitted that checking bloodpressures etc is not something they would automatically do for EDS sufferers, but would if asked or symptoms presented. I would ask them to, and ask them to tell you what is the normal range for a child of your dd1's age and size. Now my Rheumatologist said that there is medication available to counteract this, something that is regularly used in Germany, but not in this country as there are concerns that it may damage the heart, no evidence though. I wouldn't think that you necessarily what to medicate her for this, but I do think that you want to know what is happening, and you like me are cheesed off with medics fobbing you off becasue they do not know what they are talking about.
Remind me are you seeing a rheuamtologist at GOSH, if so how about writing to them directly and ask them to see your dd1 to confirm that this is a related EDS condition, then they can monitor it, and provide you with some management advice. How does that sound to you, I can't stand the idea of you stressing over what is going on, I know how that feels. Does this help, plaese feel free to tell me to shut up, no offense will be taken 
My ds and I both get severe joint pain if we have a virus or a fever....what is that about? No-one has ever been able to tell me, but it is like any virus that takes hold causes inflamation in the joints, affecting the most hypermobile ones the wors and it is agony, does that sound familar to you Ariane, or anyone else? I have to say I am guilty of doing exactly what your ds does by lying on pillows to stretch the spine back the other way, it really helps me, I can actually feel the vertebrae separating ....sorry too much gross infor 
When I was organsing the MRI for my ds I spoke to the mri supervisor directly, and he contacted the anaesthetist for me. They said that sedation normally did not work for children in these circumstances, and they describes exactly what happened to your ds. They said ga was the only way to go, but they can adjust mixtures that are used to get a light ga. It must be possible to get the medications adjusted for your ds's allergies, it is a matter of them knowing, and you being confident that they know. I know they had to be ready to adjust medications for me when they were worried I would need a cs for dd1, as I have an anaphaltic allergy to NSAIDs which are always included in the drug "cocktail" they give surgery patients. Try and put your mind at ease as much as possible about the ga.
Have I helped or should I shut up now
, I hope you are all right.