Copper wire wrap as tube damper - heat sink?

From your description, it will neither provide any damping nor any significant heat dissipation. Next.

How can copper wire damp vibrations in the glass?

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..