I think that the fascia is a red herring. There is no need to postulate a soundboard or the need for a stiffer fixing. As the OP says, the sound could simply be conducted through the fairly rigid materials of the building. (The air in the room might resonate, in which case there might be louder and quieter positions in the room.)
The thing to focus on is the source of the sound which is the violent vibration of the cable, which must be caused by the wind. I wouldn't know the mechanism, but there is presumably some consistent effect (e.g. the formation of vortices which exert a periodic suction on the cable). One needs to seek a way to disrupt this response (since you can't stop the wind). Ways of doing this depend on the vibration mode, which as it is visible may be transverse vibrations. Changing the tension in the cable might make a difference, as might fixing a weight to the cable some distance out from the fixing. Or mounting the fixing on a spring and dashpot (like a shock absorber) to take the energy out of the vibration.
It will be interesting to hear if there is a 'proper' solution.
In case anyone hasn't seen the Tacoma Narrows film showing an effect of wind-induced vibration, here is one version https://www.youtube.com/watch?v=j-zczJXSxnw