I'm no EM expert, but is it possible to have a transparent Faraday cage material that lets a capacitive touch screen register touches and be seen without any leak of radiation/data? As I understand it a big conductive finger crossing a Faraday cage breaks it quite completely, but I'm not certain of this.
No because your finger could operate as an antenna. You're conductive.
Probably? Transparent -- ITO on glass is the usual answer. You can deposit the ITO in patterns rather than as a continuous layer; and a conductive layer works as a faraday cage as long as the gaps are significantly smaller than the relevant wavelength (so ~mm for mmWave). So a ~500 µm grid could be laid down on glass in front of the screen, and conductively joined to a continuously conductive layer surrounding the back of the phone. The question, then, is whether the change in capacitance from a finger is observable by the touch screen through such a mesh... my intuition is that it would be, but would have to either model it or test it to find out. (Could test with just a stainless steel mesh from the hardware store.)
But none of this helps with the "toggle-able" part of the requirements…