Is there a budget-friendly product superficially similar to the 316 mesh or perforated steel used in high end security doors? I'm not after the same level of security, but I like the general aesthetic, the almost-invisible effect, and the weight/substance. It's for an outdoor screen room that needs to be pet-safe (cats). I have used petmesh for similar applications, and that will be what I use if I don't find anything like this, but I think something with that little bit of extra substance will let me cover wider spans with fewer spacing poles.
Are you able to share a photo of the room? It might help @MitchellMc and @redracer01 to be able to suggest solutions if we can have a better idea of the dimensions of the space. We're lucky to have lots of pet loving community members who might be able to offer some cat friendly pet safe advice too.
I've had an extensive look at all out suppliers that deal with mesh products. Unfortunately, I haven't been able to find any products in-store or off-range that match the description of what you are looking for.
Have you considered using shade cloth? I drove past a property which had enclosed their outdoor room by stretching it tightly over timber frames. It looked quite pleasing.
Let me mention @ TedBear to see if he can think of any alternatives.