This is a neat counterexample, thanks. The only thing I do not understand is why $C_x$ is decomposable in a free and a torsion part. But one could also argue that the assumption that T is triangulated leads to the contradiction that Z decomposes into a nontrivial direct sum of abelian groups. The situation you mention in the end handles Z[1/n]. I think this even works without the assumption that T is tensor triangulated - the morphisms which become isomorphisms after multiplication with some power of n form a system which arises from a cohomological functor (Weibel, 10.4)