Without knowing exactly how the screen fits into the window, I'm not sure that this will work, but here's what I did (I had exactly the same problem - the cats learned that they could get out by pushing against the screens.)
In my case, the screens had an aluminum frame - I took a few small finishing nails, and gently hammered then through the frame, into the channel in the window frame. I made sure that there was a bit of play in the nail so that they could be pulled out in an emergency (some of the windows were in bedrooms).
Don't get to hung up on the exact size the piece of screen needs to be. simply measure the width and height of the frame and buy a piece of screen large enough. Normally it comes in standard widths and cut to length. Once you lay and stretch the screen over your frame, install the spline on one end, stretch, install sides and opposite end. An inexpensive spline roller comes in handy here. Use some sharp scissors, utility knife or shears and trim off the excess close to the spline. Done!
Best Answer
It’s called a “wicket”. It’s used when the window screen is on the inside and the window has an operator (crank, etc.) that opens the window “out”.