"Has anyone ever tried a short piece of shrink-sleeve - "

Probably not a good idea. It will help the sealing, but the contraction when heated does not impart a lot of actual compressive force. Also, you will need a razor knife to get it off, which will likely damage the tubing.

I like these:

https://www.amazon.com/Wingsmoto-Tub.../dp/B012MBHEHQ

8.5 mm may not be exactly the size you need.