There is a product called PET SCREEN, we have this installed in the lower portions of out lanai and birdcage. It is a much heavier designed screen and is supposed to withstand a 20 pound (or more?) cat climbing it with no damage. There is also a few companies that do screen replacement, so you should be able to have the existing screens replaced with pet screen.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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