I think this is clinically quite a challenging area and would agree if you aren’t experienced in the diagnosis of EDS then you should probably refer to a rheumatologist or intermediate service (whichever is the pathway for your area). Some require referral to genetics depending on additional features (although in my region the genetics department aren’t actually very interested!). There may also be overlap with other causes of chronic pain which can complicate the clinical presentation.
Just for information the EDS website is good to learn more about the 2017 diagnostic criteria
Generally blood tests aren’t helpful unless you are looking at an additional chronic pain type picture in addition in which case maybe vitamin D levels or thyroid maybe useful.
I’m not sure what the diagnosis confers in terms of access to benefits. Someone else maybe able to advise on this.