i wish i had ideas for you...i just wanted to say that it's really sweet you are thinking of your kitty kats :)

These doors in particular don't have screens. Screens simply don't work for doors designed to open as these do. On other projects with large sliding doors we've used a roller screen which attaches to the jamb of the door on one side and extends across the opening, connecting magnetically to the other jamb. This way the screen is out of the way and not seen when not needed or wanted. Perhaps a roller screen could work with these doors if you create a very deep jamb and head. This would allow you to "crack" the doors open for air, but keep the bugs out, or in your case, keep the cat in.

Phantom screens

embed a roller screen that pulls down in the header over the door between the pivot points of the doors, so that they will come down when the doors are open

Mirage screens are the best choice, will hold up better than any other screen on the market and have a pet mesh option that is more durable than any other competitor screens, good if you have cats... I have used Mirage screens for all my happy clients.