Alicia: You can try to find a retractable white shade to put the sign on and put it over the water/ice dispenser too. (I thought the same thing you did.) That way people will usually see the sign, but can still access the ice and water (which is the best part of those fridges to me.)
Never thought of that one, but since my son already has TARDIS bedroom doors I think that might be the next project. After all I already have the oxford blue paint.