Your principle TC implies the rigid relation principle RR, a weak choice principle introduced by Justin Palumbo and myself in this paper:
- Joel David Hamkins and Justin Palumbo, The rigid relation principle, a new weak choice principle, Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:1106.4635.
The rigid relation principle is the assertion that every set admits a rigid binary relation. This is true under TC, since $\langle X,\in\rangle$ is rigid whenever $X$ is a transitive set, and so every set that is bijective with a transitive set admits a rigid binary relation.
Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.