I maintain that from the time he requested permission to review the documents, he was being watched. I also surmise that special steps were taken (like changing which employees observed and handled the video tape) so that Berger couldn't get any inside help. Perhaps the "inside help" decided national security in a time of war was more important than covering up for the Clintonistas.
Did you read 624?
You're right, Berger thought that they had the Archives buttoned up. But if that's true, why did he stuff the docs into his clothing? Presumably, he could just stuff his briefcase and walk out.....