The staffers run afoul of the law only if they get stopped. Congressmen themselves are covered under Article I, Section 6: "They shall in all cases, except treason, felony and breach of the peace, be privileged from arrest during their attendance at the session of their respective Houses, and in going to and returning from the same..."
The problem isn't D.C., it's when they get to work. By the laws that Congress itself has in place nobody, be they Congressman, Senator, or staffer, can bring a firearm into the Capitol or office building. They are defenseless at work by their own choosing.