MASSIVELY LONG REPLY:
also, if her heart rate rises on standing, look at PoTS (postural tachycardia syndrome) which is linked to EDS. The PoTS can be managed, although with her other health issues that would be trickier.
DD2 has (mild) EDS and (not mild) PoTS - although her cardiologist seems to have changed her EDS diagnosis to benign JHS, which I'm not too impressed with. DD1 has JHS, although she's affected more by the hypermobility than DD2 is.
It seems that rheumatologists disagree as to whether they are the are the same thing, or just very very similar. Plus some of the other variants of EDS are much worse than hypermobility type (was type 3). But, treatment ought to be the same regardless of the diagnosis, as it's the symptoms which are treated.
Also linked with EDS/PoTS is gastroparesis (literally "stomach paralysis"), so after an hour and a half or so, food comes back up, as the digestive system has shut down - DD2 spent a long time being told her was due to constipation, and it took MH input to finally get somewhere. But yes, it's rubbish when you know your DC is actually ill, and because they are a teenage girl, the automatic assumption is anorexic/pregnant/on drugs/makign it up for attention.
Don't know where you are, but there is a rehab programme at Stanmore (near London) for teens with EDS, very strict criteria for admission, and long, long waiting lists, but I hear very good.
If you can afford a private consultation (I think he's around £350) then the person you want is Professor Rodney Grahame in London. For NHS, then Hypermobility.org have this small list.
Otherwise look at the website of your local hospitals and see what is listed for the interests of the rheumatologists there, as there might be a hypermobility interest shown (although hypermobility gets put to rheumys as it doesn't really fit in anywhere). You might find some specialist physios locally, again, really really quiz then hard about their expertise, as your DD has pretty complex issues. Jane Simmonds at St Johns and St Elizabeths (the Hypermobility Unit above) is v good with EDS I understand (DD saw a different physio there who was OK but not amazing).
To manage the PoTS you're looking at a cardiologist probably - I can recommend one in London, but if you can say what area you're in, I can see if there's anyone local to you - I'm on a FB group for parents of DC with PoTS so we have recommendations country-wide.
To summarise, if it's EDS hypermobilty you think she has, getting the diagnosis changed probably won't make much difference (other than some doctors think it's a more severe version, and sometimes take you more seriously). If the rheumatologist you saw didn't set up a treatment plan, or send you somewhere for that (some consultants do diagnosis only, then ask the GP to arrange treatment - Prof G did that for DDs EDs, but did refer to an amazing paediatrician for the PoTS), then you certainly need a different rheumatologist, and the GP should sort that - not for a new/different/correct diagnosis, but for some treatment.
Finally, I spoke to DD while writing this mammoth post, and she said to tell your DD to try counselling again if she can - DDs paediatrician explained to her that generally the symptoms are a sign that you body isn't happy with what you're doing to it (in DDs case, not enough water and regular exercise, and falling out of her routines), so if you're stressed about eg being ill and not getting any treatment, that can very easily show as gastroparesis (or in DDs case fainting). Getting things sorted in your head (CBT can be very very useful for this I understand) will help the symptoms. That's not to say that it's all "in the head", but getting your head in the right place helps an awful lot. We were very close to psychiatric inpatient admission before DD got a medical diagnosis because of what her psychiatrist called "an extreme but normal reaction" to her situation. Knowing what was wrong, how to manage it day to day, and getting your head round the whole situation makes a massive difference - DD went from collapsing 120+ times a day to (mostly) managing by herself at uni five years later.
You have all my sympathy - please PM if I can be of any help, particularly if you don't want to put your location here 