Carbon tubes are very tough, EXCEPT when subjected to compression forces. Some bike shops have workstands that clamp to the frame of the bike...excessive clamping forces can shatter a carbon tube. The same thing has occurred with some vehicle racks that use clamps.
But, as long as you are not using any clamping or compression device to mount your bike to a trainer, you shouldn't have any problem.
Most Goodwill stores and Salvation Army stores have bikes for $20...cheap training bikes.