The thing is that they already have the framework to do this. A senator is immune from prosecution for anything state on the floor of the senate. If Wyden is on the senate floor, he could read off the nuclear launch codes and there's not a damn thing that the executive can do about it. Furthermore, the senate has closed sessions. They had one just a couple of years ago for the START treaty. A senator could take the floor, tell the rest of the senate about any classified material they like, and the information isn't released to the public unless the Senate votes to remove the injunction of secrecy, so the information remains secure. Even if the injunction is removed, the senator is still protected by some fairly strict constitutional privilege.

This seems to be a solved problem.