The woodpecker likes the sound, that's why it starts knocking. The plaster layer on Styrofoam is usually only a few millimeters thick and it gets through quickly. As far as I know, even fabric filler doesn't stop it. Once it's inside, it has found a great nest, right? :D Insulated all around, additionally heated from the house through a thermal bridge. That's a perfect place to hatch.
As a temporary measure, I have read so far: - extra thick plaster layer and extra thick fabric filler, together at least 1 cm - very smooth plaster, so that it has no grip
What protection device are you planning to use? I assume something else since you already mentioned a "device."
By the way, the NDR show just mentioned mineral foam boards (something like aerated concrete in the end), which supposedly do not sound hollow and are rather unattractive to woodpeckers. Is that true? I don't know. But a complete rebuild of the thermal insulation is obviously an extreme measure. Another option would be brick cladding. :D