I can believe the shape of the crook can make a difference, but the fabric?
For efficient coupling of energy from the air inside a cylindrical pipe to vibration in the pipe wall, the frequency must be greater than the "ring frequency" of the pipe wall. That's the frequency at which a ring of the pipe wall material will naturally resonate in a "breathing" mode, where the whole ring moves outward or inward in phase, as opposed to higher-order modes where some parts of the ring move out while others move in.
I've knocked up a spreadsheet at
http://sheet.zoho.com/public/dave.mclau/ring-frequency. 25 mm is, at a guess, about the largest diameter you're likely to see on a tenor crook and, assuming the pipe is brass, it results in a ring frequency that's about twice as high as a typical young person can hear! A softer material, with a lower wave speed, will reduce that frequency but, for any likely material, I don't see it coming down enough to matter.
At frequencies above the ring frequency, the material of the pipe wall can affect sound propagation inside the pipe. By flexing, it reduces the effective speed of sound inside, and hence the resonant frequencies. The pipe wall can also efficiently radiate sound outside the pipe. Below the ring frequency, I don't see how the material can significantly affect either the sound inside the pipe or the sound radiated into the far-field.
Of course the crook isn't cylindrical, or even straight, so it's not as simple as I suggest, but I don't believe the material can affect the sound. It's also possible I've misunderstood something and I'm talking nonsense!