I have a fair bit of experience "containing" barrels, but not with rimfire. With CF, heat will be an issue. With the large 1K heavy guns using large aluminum or steel tubing, we've put a water/anifreeze mix in the tube. Everything stays quite cool. With the LGs & no coolant: I had a 6 Ackley that for 10 shots or so would shoot lights-out. So five shots for sighters, a 5 shot group, and odds are you're in the winners circle. If you took more sighters, maybe not.
I tried that with a 10.5 pound PPC, and put water between the tube & barrel. But the tube was so close to the barrel at the back that the water wasn't even. That barrel setup would string shots down for the first 5 or so, then back up (still stringing) for the next 5 or so, before settling down. Needless to say, that wont work. I took the water out & things were better; there isn't as much heat from a PPC. I used that rifle for a full 2-day match, and had no issues with heat. It also didn't shoot too well, bottom of the pack. Whether that was me, the effect of the tube, the build, or a so-so barrel, we'll never know.
Moral: It may work, it may not, and don't forget heat, even in a RF. I think you'd have better luck with a tensioning or compression tube, but this is worth doing too.
Thank for pushing thing forward,
Charles
Edit: There have been some hints that a carbon fiber tube might help with vibration. Apparently some properties of carbon fiber. Of course, heat will build up; unlike the metals, carbon fiber is lousy at passing heat. The 54:1 issue of PS featured a cover photo of a .338 with a carbon fiber outer that had, the builder claimed, reduced recoil, because the Carbon fiber barrel. The builder was David Smith, who came up with the system marketed by Magnum Research. He claims "The carbon fiber suppresses vibration, but also reduces recoil by minimizing barrel flex so it doesn't snap back..."
Not that we care about recoil, but the other properties are interesting.