I used to have a dog who always tried to chase anything that moves. To keep it from going through the screen, I bought one of those screen door protectors readily available at any home improvement stores. They come in several sizes. I bought the smallest one and modified it a little to fit it against my door. That was on my old travel trailer, otherwise I would put a picture up here.
Luckily, my current dog is not much interested in stuff moving outside of the motorhome, and she respects the closed screen door. With about 100+ lb of weight, she could pop the door open anyway.