Hi Ben,
Are you in SW London? This is a job much better done with help. It's not a big job but it is rather important. I would also like to see the failure.
The ring is a seasure one I think. You want a small overall diameter and a big rod thickness. I think it was 20mm od and 4mm rod. Measure the other one.
The ring was bogged on to the rack, and then in a single mix more bog is put over so there are minimal/no hollows. Then individual strands from carbon cloth are wound round and round, having been wet out already. It is not that important to put on loads of fibres because they are very strong, but it is important that they are spread out as they go round the tube. The other thing that is important is that the strength in the art which goes through the rig is good right up to the edge of the strands. If it tapers down then slight flex in the rig can cause the carbon to break one strand at a time.
The individual strands are called 'tow', pronounced "tau" or "toe" variously.
People do use strips of uni, but we used tow. With regard to how much to put on, in section carbon is stronger than steel (which is what the shroud is made from ), so as long as there is more than that in cross section, (taking into account mechanical disadvantages etc), then you'll be fine.
I think we used huge tow from a roll of 'mega-weave' which is quite good for making formula 1 cars, but is too heavy for making cherubs from. There is probably some still around. It was a gift to the class from a formula 1 team and has been used for all kings of funny jobs, like rudder stocks, shroud points, fronting plates, etc.
We can get together to fix this at some point.