I think the furniture pads will be artificial fibre, Clive. Burn a bit of the fibre, and see if it smells like burning wool. You could do with with an old felt hat to cut up.
I wonder if felt strips could be wound round the wooden taper, like a fairly coarse taper thread. Turn the thing while gently inserting it, then pull it straight out. It might have more chance of dragging out chips that way than if the strips were longitudinal.
Andy