Never tried to repair Carbon Fibre so I'm short of ideas. I think it's too difficult. Carbon fibre rod is made of extremely thin carbon fibres embedded in a brittle resin matrix. Rod is very strong along the fibres, much less sideways. This picture is of the suspension of my experimental clock pendulum, where I removed a section of matrix so the pendulum only swung on the flexible fibres, independent of the resin. The example is 0.8mm diameter, but the construction is the same in bigger diameters.

Appearance depends on the matrix, which should be easy enough to recreate. Unfortunately, the strength of the rod depends on the fibres, which can't be reconnected. The best I can think of is like Clive's answer, except I'd use a steel rod rather than Aluminium tube. (Carbon fibre is stronger than steel!)
I found cutting thin rod no problem with a wire-cutter or rolling a sharp Box Knife over it, but it gets harder to do it neatly as the rod diameter increases. Radio Control types recommend rolling the rod on a wet cloth with a razor blade. The problem is the brittle matrix tends to crack and chip, causing nearby fibres to stick out in an untidy fuzz. A Dremel cutting disc might do a better job: I've not tried it!
My feeling is the item can be made to look good again, but the repair won't be strong enough to allow the baton to be waved about.
How about the dead Hamster fix? When the child's favourite pet dies, take the corpse to a pet shop secretly and replace it with a near match. If all else fails, tell the friend to substitute a new one without telling his wife!
Dave