Not only will wheelchair services not provide a chair if you don't need it outside the house... but if you DO need it inside the house but cannot use one because the house layout prevents it... you still get turned down.
However they DO typically provide them for children to access education, so that seems a little odd.
You are right that self propelling a badly fitted wheelchair is bad for you, actually self propelling even a well fitted well balanced chair is still bad for you - my rotator cuffs are knackered and I now have constant pain from them...
However you are wrong to assume wheelchair services a/ are the only ones who can correctly fit and balance a chair or that b/ they would do so.
Typically they will take measurements, and supply the nearest chair that fits that person - they rarely look at HOW someone moves or sits in any detail. Often the chair will just be delivered to you and its up to you to adjust it (if it can be adjusted).
I have found that custom wheelchair specialists (mines an RGK) are FAR better at measuring and fitting chairs and ensuring they are set up correctly for you.
With hypermobility such as EDS, you are damned if you do, damned if you don't really..
Don't use a chair, damage your over stretched muscles and lax tendons/ligaments because you move incorrectly, and fall..
Do use a chair, bugger your shoulders and back using them in ways they were not designed to move.
For a kid however I think on balance, using a chair that will likely be mostly pushed by an adult, is going to preserve joints longer than not doing so.