Micro mini blinds like these from
justblinds.com are 1/2" wide and will fit between the screen and door.
The thing about putting anything between the screen and door is the hassle factor of opening and closing whatever you put over the glass.
Another option is to buy a "lambrequin" (the fabric covered valance at top of a window) for your brand of TT from the dealer. Get one slightly wider than the door and modify to fit over the door. Then use any type of blinds which could be one to match what's on the windows now.
I like the 1/2" micro mini blinds idea, but since DW is a bean-counting accountant, we have the foil insulation stuff mentioned above in a precut/trimmed piece. :( But at least the early morning sun won't shine through the door window like a searchlight anymore.