Before doing anything permanent, consider a little strip of electrical tape. If electrical tape isn't thin enough, Kapton tape is cheap and readily available (and useful!) and is incredibly thin. It might help.

Part of it is resonance throughout the entire metal body components of the knife, as well. A bell doesn't ring just because the clapper hits it, it rings because the whole surface is allowed to vibrate afterwards. So consider adding material to dampen not the contact surfaces themselves, but any large, flat, and hidden surfaces that are -- most importantly -- free of any mechanical interference. This may be easier.

You probably won't find anything marketed as sound insulation that's thin enough, to look into finding generic self adhesive rubber sheet or felt that's thin enough to meet your requirements. I have a knockoff M, and I'm sure there's about 3/16" of clearance around most of the interior of the blade track, possibly excepting the part immediately left and right of the spine.