You have no right to do any upgrade or modification with his written permit.
Think this, the rent price was based on the unit that showed to you when you and the landlord view it together. You and the landlord both agreed at that time on the unit condition and the rent price.
There always some tenants ask for upgrade after move in, like the kitchen cabinets are too old, need to be replaced, Washer need to be replaced by a Front Loading model.
I think your landlord will not do the screen, even this is the small request. because he does not know your personalities, and so far seems you do not understand your boundary on what to ask, he might suspect that you may have a long list of demands later.
The best way for you to get the screen is to offer to the landlord, that you will install a screen at your expense, and tell him how you plan to do it as good as a professional. When he sees that you do understand what you are doing, and there will be no damage to the windows, he may agree.