the shorter the "lever," the smaller the required motor torque (but the door won't open as far with a smaller lever, either)
I think we are agreeing. At the moment the lever is working against the motor, so the shorter the lever the less torque the motor needs.
The last set of photos seems workable.
Would it not be simpler though to tie a good thick piece of cord to the handle and just train the dog to pull it?