You may have worked this out by now, but If I understand correctly, I think my setup does what you want.
Go to menu SHOOT7, "Shooting info display". Screen info settings. Make sure you have #6 checked, this is "screen off". I have only one other option checked so that the info button becomes an on/off toggle for the screen when you are shooting. But it doesn't need to be a toggle, you can have several shooting info screens along with "screen off".
When I'm shooting, I usually have the screen open & folded into the body, and toggled off with the info button. It then stays off when I pull back from the viewfinder. I can switch it on either by hitting the info button to show the shooting info screen, or by hitting Menu or Play.
For some reason #6 "screen off" in that menu is only available for still shooting.