Stainless steel insect screen (sometimes referred to as security screen) is available but it may not work in your existing screen frames. Just google the phrase "stainless steel insect screen" (without the quotes) and you will get a lot of hits.

The high visibility screening materials out there (Pella calls it Vivid View while Andersen calls it something else) is a stainless steel mesh.

It is going to be sturdier than fiberglass but much, much more expensive. I am not sure where you can get it in rolls whereas fiberglass is very easy to find.

I think you will be better off with fiberglass and a screening tool.

Another option is aluminum screen mesh. It is readily available. Slightly more cost than fiberglass, but more durable.
You may need a smaller diameter screen spline. This type is a little trickier to roll into the frames.
I had a small dog that used to like to make his own doors. This solved my problem.