I would suggest that 7700 would be a good idea.
At the unit I work, we work a bit of Londons Airspace and we wouldn't get any of their emergency flags transferred onto our system, and, if they are busy moving things out of the way, we might not get too fast a phone call!
I would suggest setting the A7700 then we will have you flashing (ooo errrr) on our screen, a message on our screen saying special code and our attention would be drawn to you giving us as much notice as possible, if both things happen, then its even better IMO.
Maybe as emergency descents are so rapid they could warrant their own SSR code?
TIO