From your description, it will neither provide any damping nor any significant heat dissipation. Next.
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I use a metal ring purchased from the craft store and adjust to the proper size to fit with 3 separate black plumber used faucet washers in 1/4" diagmeter. Idea is based on the Herbie O-HAL dampers. It works pretty good.
For bigger tubes like 6C33 or KT88, I cut a piece of goat hanger and shape it in circle and fit with bigger washers. It works very well too..
A soft copper alloy could indeed provide some damping. However, it won't help with thermal issues. A tube uses radiation, not conduction, to remove heat. Think of it as a light bulb. Anything surrounding it gets lit - with heat. So you actually do NOT want to trap the heat inside by reflecting it back or by capturing it right at the glass surface. So let the light out (technically invisible to the human eye).
If you do use copper, put it at the top where the getter flash is.
Of course, you now introduce other issues. One, you've created an antenna. And depending on how it is wound, a transformer secondary winding. What might happen is coupling of stray and unwanted signals into the circuit. It might can some RF. It might react with a stray magnetic field from a nearby power transformer. Sometimes just grounding the wire might be enough.
And you'll have to find some way to wrap it tight. I'm thinking there is a bit of thermal expansion with Cu, so if you wind it tight while at working temperature, it may end up shrinking too much and breaking glass when cooled. This is doubtful because of the softness of the metal. More likely it will just work its way loose.
Such is the way of progress. What seems like a simple idea at first, may actually end up very complicated in practice. This might be one of those cases of 1 step forward, 2 back.