Screen materials come in very many mesh sizes. Consider screens other than window screen material. The finest I have seen were brass or stainless steel. Do a google search for brass screen and stainless screen. They are used in filtering. Places like Small Parts and other similar industrial supply places are good places to start. Some of the better hardware stores have a few sizes of brass screens. Paint stores may have some filters that have usable screens.
I find brass screening material much easier to work with than stainless, if you have a choice. I find working with stainless screen I poke myself a lot and end up with tiny holes in my fingers :-(
I have also seen photo etch grating material (not screen). I think it may have been K & S but not sure. I have seen them in hobby shops, and also a couple of model ship supply places like Bluejacket or Model Expo.
Michaels and Hobby lobby have fine chain material in their jewelry craft sections.