I have H-EDS. Finally diagnosed in my forties but have had recurrent dislocations and big surgeries for it since I was 9. Seemingly doctors just didn't know what it was and I was "just bendy".
The lack of management and taken its toll however and I now have osteoarthritis in all of my joints and need some joint replacements.
I've also recently been diagnosed with dysautonomia and I have orthostatic intolerance and reactive hypoglycemia. Again, all linked to H-EDS.
I am lucky that the hospital where I'm treated for inflammatory arthritis is also a specialist centre for hypermobility and they have a programme for it which includes Pilates, gym sessions and hydrotherapy. We were encouraged to pick and mix and then to continue doing what we enjoyed. Going to the gym was encouraged. Using weights was also encouraged but focus was very much on form and using joints correctly. So, try and face a mirror so that you can visually check joint placement as proprioception is affected by hypermobility. We were also encouraged to not move to the end of our range whilst in the gym and to use machines rather than free weights because machines tend to help you use joints correctly.
In Pilates it was again encouraged that we looked at ourselves so that we could see how our joints were moving, where they were placed and not move to end of range.
I love Pilates now but do still have to be able to see myself in a mirror. When I do physio I have to do it in front of a mirror too. I also do yoga which isn't the best for us bendies but I enjoy it and it helps the pain. The drs are quite relaxed about it though. It's not a definite no no.
Have a look at the HMSA website. That's really helpful and also the ehlers danlos association site. Jennie de Bonne on YouTube also has some really good exercise videos.