Then run a simple optical fiber cable for communication to an antenna that lies outside the cage.
If you are really good you have multiple antennas that are kept off (with the circuit physically disconnected from the antenna). If you get hit with a pulse activate the next antenna.
It's possible to make simple electronics that can survive such a pulse, with auto-reset fusible links, and you use those for activating the antenna.
Or even a small door and "push" the antenna out of the Faraday cage when needed.