How can I keep a door open wide enough for my cat to get through but not my big dog?
It may seem too simple to work, but you just need to cut your own tailor-made cat door. 

Basically, a cat can get through any opening it can fit its head through. This usually works against us cat owners, since the beasties have a mind of their own and like to explore. This time, it may work in our favor. So: 


Measure cat head (breadth), add 1/4 inch to make things easier.
Cut opening in door about that width, of reasonable height (8" should do the trick, perhaps even less).
Teach cat to go through the door, perhaps with some treats on the other side.
Watch dog try to fit her wide head through the opening, fail miserably. 
Watch dog proceed to try to make opening wider by scratching.
Patch up the sides of the opening with metal.


Hope this helps (the cat).