Add a pet door to your screen door. 

Indoor and outdoor frames sandwich the screen and are secured with M3x14 screws and nuts. The swinging hinge is a set of captured M3x40 screws and nuts. Once in place, cut the screen around the inner frame with utility knife to free the swinging portion of the door. Optionally you can glue some magnets in the insets in the door to help keep it closed though I haven't found it necessary to do so.

The model is (I think) fully parametric so you can adjust to the size of your pet and print bed. I printed these all flat but I think you could probably get away with printing them vertically…