The main reason not to bring a phone into the room is that the phone could be compromised. If the person is compromised then a device isn't your problem, because they could view the documents and copy them on paper or just remember the contents to write down later.
The yearly limit for rad workers is 5000 mrem with most receiving none. Receiving any dose is usually a cause for concern at most facilities that handle radioactive materials. A full body x-ray would dose you with about 1000 mrem. For about every 10000 mrem you receive, you gain an additional 1% chance of lifetime cancer risk. There's a reason why you wear a lead apron when getting X-rays at the doctor's office and why the technician leaves the room.
Metal detectors would be a much more reasonable method. People that work at airports, courts, jails, some schools, and even some manufacturing facilities walk through metal detectors daily.