re: People with experience installing screens on a patio

How about installing the screen on the outside of the wooden structure you will use to help close in the area. Then on the inside get some of that 1/2 x 1/2 inch mesh wire and run that up about 3-4 ft. and secure that to the inside of the wood framing. That way the pets cant' get to the screen to tear it up. If using 2 x 4 for framing, there will be a 3 1/2 inch space between the inner mesh and the outer screen.

It would work better for the dog, but cats love to climb and that may present an issue in the long run.