The idea that the actress playing the refugee child wouldn't tell the world what happened simply because she signed a contract is frankly unconvincing.
Also the very fact of casting an aspiring actress seems like a pretty huge risk, especially if she hits it big afterward.
Wasn't she told that if she goes public, the secret service would have her killed?
For that matter, the entire United States Armed Forces are not in on it, nor are any of the inevitable swarm of reporters sent to war zones. Or the entire nation of Albania and its neighbors. Or any other country with any kind of contact with it. In fact, the ruse can only work if no-one in the media does any work whatsoever other than read press releases and run stock footage.
That's the point—that the public will accept anything the media tells them, the media are nothing but puppets, and the puppet masters control everything.
Remember Baghdad Bob? Would it be too far fetched imagining the US media ridiculing an Albanian spokesperson, when they can show what's really going on in Albania?
Or the entire rest of the world could just go to Albania, notice that the only thing remotely dangerous would be the possibility of your car getting stolen, go back and report it and have a very long list of powerful people asking uncomfortable questions to their American counterparts.
Wag the Dog is based on the book American Hero by Larry Beinhart, in which the President is, specifically, George HW Bush.