If I was needing to use a carbon tube in a linear bearing, I would consider bonding a steel sleeve over the shaft for the bearing to run on.
The coefficients of linear thermal expansion vary enormously. Plastics expand and contract a lot more than metals, so in cold weather that steel sleeve might become debonded from the composite shaft.
I always found it better to design machinery that had pivoting parts rather than sliding parts. Sliding stuff is difficult to seal, so if you have close tolerances, any bit of grit can cause galling and seizure. Larger clearances lead to rattling and slop. Friction is almost always higher with sliding stuff.
The Cessna 400/ttX/Corvalis uses aluminum rods running through guides that are nothing more than three small nylon wheels. Even there they have to have clearances, especially the shaft that runs immediately aft of the stick, since it both rotates and slides. There was a service bulletin on the aileron pushrod guides; they were chafing the tube. Again, vibration and grit cause trouble. Cessna also used the three-tiny-wheel thing in the 210 and 182, in the control column glides, and rattling, chafing, and wheel failure was common. The wheels were small ball bearings that had nylon "tires" molded onto them, and wear, age and vibration would break the tires, especially if they were set too tight (rotating the shaft put high side loads on the tire, breaking it) or too loose (vibration and rattling would wear or break it). They would have been better off using a simple plastic bushing like the 150/172/180/185 and so on, but since the heavier airplanes have higher control forces, and the pilot tends to pull down as well as back in the flare, for instance, friction would be higher.
Some homebuilts as well as certified airplanes used pivoting arms to support pushrods. Avid/Kitfox and the Lake Amphib are examples. Totally pivoting systems that have very little friction and are relatively free of binding. The arm is a triangular affair, with a long bushing along its base that stabilizes it sideways, and the pushrod mounted to the pointy end. The pushrods are mounted close to joints with further rods to enable directional changes. The Lake has, IIRC, three or four of those pivots in its elevator system, but you can't feel them at all. Rudder is similar.