I think the answers are obvious; as much as we would rather conceal the truth, it will be difficult to do. Especially given the size of the displaced population, and the number of us who are "fictional" or have duplicate faces, or are otherwise significantly "abnormal" for this place and time.
However, having someone - or several someones - dedicated to the "damage control" side of things would be ideal, whichever direction our story is spun.
If the military - or anyone similar - attempt to do anything of that nature, I suspect they will have a difficult time. Still, you are right to be concerned.