The safest thing to do in my opinion is to get very thick, good quality timber and to double mesh on the inside. Foxes and other predators find it harder to push mesh off when it's attached on the opposite side than they do to pull it towards them, as they can't get their teeth under the mesh and pull. One layer of mesh should be 1/2 x 1/2 inch galvanised weld mesh and 19g (or 16g or better), and the other can be whatever you fancy - perhaps a very thick and large mesh might be good if you're very concerned? Two layers of the same mesh should be very secure though. The mesh should be nailed with u-nails as well as the staples that it might come pre-attached with. The more u-nails and staples the better.
Make sure to have big bolts to any external doors, especially at the bottom. Predators are generally at floor level so you need a bolt there. You can attach caribiner clips to the bolts too which should make them pretty fox-proof, although I'm not sure how possible that'd be with your disability? I got the 60mm one from here and it's very easy to open -
http://www.ebay.co.uk/itm/316-STAIN...mbing_Mountaineering&var=&hash=item27d35b0367
Additionally you could screw an external kickboard of timber around 1/2 or 1 ft tall around the outside of the aviary or/and screw timber to the inside to clamp the mesh in place at the bottom as that's where it's most at risk.
A bright security light aimed at the run would also be a good idea, but try to place it so that the bunnies don't set it off all the time!
I'd also have the 'door' between aviary and shed as a runaround standard size door. They're designed so that a fox can't fit through, so it'll give an extra layer of protection. A double door will allow you to shut it off from either side which is handy.
Personally I wouldn't bother with all of these, it's just a load of ideas for you. Personally I'd probably just go for the strong bolts + clips definitely, the runaround door and the security light; and the double mesh if you're very concerned.
I would definitely go for a wooden timber construction over a metal aviary unless you're able to get the metal aviary in 1/2 x 1/2 inch mesh - the risk from cat scratches are not worth the risk. You could get the metal aviary/dog run and mesh over it with smaller mesh though, if you feel that'd be better.