LOL Love your story..
Well I think we can work something out here because I have birds that keep coming in through Peanut's doggy door.. so if you could you can send me that kitty to get the birds it would be a win- win
I did a quick Google search and wasn't able to find anything, however there are these magnet collars for the doggy door.. I'm sure you would have to buy a whole new doggy door- I guess the dog’s where the collars with the magnets and the door will open and close.. That sounds like it would scare most Yorkies.. I know Peanut would be terrified if it opened by itself!!
Sorry I’m no help.. I would honestly be blunt with this lady and say if your cat comes in my house again I will take him to the shelter ( you obviously don’t, just to scare her)