I found the battery completely drained after 10 days in the garage. It's on an AC charger now.

The trip odometer and odometer light up when you open the driver's door. I think that's normal. Several seconds after you close the door (from outside the car) the two displays go blank. But they remain illuminated! They stay that way. That can't be normal, and it could explain the dead battery.

What could be causing this bizarre fault? I know my way around a wiring diagram, but this is just weird.

Do your outside lights come on?
When I pull my slk in the garage on darker dimmer days the lights stay on for about a minute then they go off.
My lights do not come on, when the day is brighter even though it is a bit darker in my garage.

I would search for the timer that controls the interior lights it may not be timing out, like it should.

You might also try popping out one fuse at a time, until the odometer light go's off.
you may get lucky and all the fuse controls is the interior lights.

Good luck