I don't know how the camera knows when the LCD is swung out/open or not, but perhaps there's a switch that tells it so (and turns on the LCD). And that switch could be the source of your problem. I didn't see anything on the parts diagrams so you'll probably have to call Sony ServicesPLUS and ask 'em.
1-800-538-7550 is the parts number, they have lots of different contact numbers on that site for different reasons. My suggestion is to call the parts guy and have him figure out how that LCD panel is activated when swung out. From there maybe you'll have to get transferred to service if you can't perform the repair yourself.
In my experience, some parts do NOT appear on Sony's schematics. I ran into that when one of our part-timers let one of our EX cameras (EX1) do a backflip off the tripod last month, shattering the EVF on the back of the camera. Some of the guts I needed to make the repairs did not appear on the parts diagram. A quick call to Sony had the right part number for me in less than five minutes. Then they put out a service bulletin (which showed up in my email that afternoon) to let people know of the part number.