I've wrapped the heads of the tubes in duct tape, as its strong and silver to match the chucks. If you do it neatly it does not look all that bad. And whilst I've not used it much, it seems much better, as without it the paracord was fraying after barely 20mins spinning.
But the new chucks use 19mm chrome pipes, so I got a lighter, smaller link chain, and that has changed the dynamics of the chucks considerably. I can now do moves, like wrapping the chain round the finger to close the chucks, which I could not do with a larger chain and heavier chucks. So I may not use cord at all, as a finer chain with lighter chucks seems to be the answer I was looking for.