You should first duplicate the mesh group in WS (you should have one shadow group, and two mesh groups), so that with two groups one will be the solid parts and the other will be the transparent parts. By having the transparent part in a separate group, it will prevent the solid group from looking "inside out". Then you will need to change the material on the transparent group from Phong to Phong Alpha. When asked if you want to update the property list, click no.
Then you need to adjust the alpha mask threshold property. The lower this number is, the more "solid" your grid will appear when you zoom out. The higher it is, the more invisible it will be when you zoom out. For your particular texture/mesh I'd suggest trying different values between, hm, 80 and 120. That's just a rough guess, you can of course go higher or lower depending on your own preferences for what looks best.