Still better than a windowsill--which is about the worst place for bonsai.I have a red maple that I grabbed from a work site before the mowers got it plan to rehome it outside on the dirt if it survives, 2 ficus and a chinese elm. Unfortunately my only outdoor option is a covered northern facing balcony.
Elm can go out now
Ficus can go out once temps stay above 50 F most nights of the week.
In winter, the elm can be left on the balcony set in a cheap styrofoam cooler. The cooler will slow the swings in temperature. BUT your outdoor storage closet is a very good alternative too. Either will work.
I don't think the screen affects the amount of light entering the window and your bonsai. Even so, you can consult with the master about it, or with the florist maybe. Although I don't think they have much effect on the light coming in. I have many colors at home, but the window is significant, and when planning it, I wanted the screens not to be too large at once. So I don't have any questions about it. Maybe it really bothers me, but I ordered from joinerysolutionssw.co.uk, and I was told that it has no effect in my case. So I think you should contact them and they will answer everything.

Some screens are even marketed by how much light they shade.Window screen blocks from 15% to 30% of the sunlight on average, Some screens block more.