The varying lighting of the room inside of the inner door will always be a serious problem when trying to see through that door. So getting a camera inside of the space between the doors would be good IF you can mount it close enough to the inner door that you will have a good view of the people in that space (they're not pressed up too close to the inner door).
But you may well be able to shoot through the inner door if you can control things just so.
Let's say you mount the camera through the drop-ceiling, directly in front of and above the inner door, shooting downward at it at a fairly steep angle. This means that what will be seen in the reflection from the inner door will be the floor directly inside of the inner door.
You might be able to get them to place a black or very dark rug on the floor in the area that the camera sees reflected in the door glass.
With a focal length that lets you see only the required area of the inner door's glass, the area of the floor seen reflected in the glass can be kept to a minimum. That setup eliminates dealing with reflections from the walls and most of the floor, which will help.
Then use the camera "
tools" you guys are considering to help with the exposure correction, etc.
This is a difficult setup, for sure!