We can rebuild it....

Having bought a set of hex keys (thanks to kaberett for the Gorilla Grip recommendation) I finally took the eBay chair apart and put it back together to suit me (well, as much as a chair two inches too narrow can be made to). Hopefully it'll no longer wheelie if I breathe in too sharply, and both the brakes now grip the tyres, rather than just one and a bit.

Took the upholstery off too, and gave it a bit of a clean. The back definitely seems to have been meant for a chair with a back twice as high, and there was so much dirt and leaf litter in the velcro I think they must have been using it on a farm! I've left the side skirts off for the moment, they were being forced outwards by the cushion I'm using, so weren't really working and I think the extra bulk at the back was making the upholstery fit worse - both they and the back had padded sleeves to go over the uprights, and you could only use one at a time, so the other ended up folded under.

The front castors still need either replacing, or a new set of tyres, but hopefully I should be able to use it now without it wheelie-ing every time I try to go up a kerb cut or wheel up a slope. Which does not help!

And hopefully Sods Law means Wheelchair Services will promptly ring me to say the new custom-built titanium chair is ready to pick up ;)