This is a partial answer, but let me mention that the transitive choice 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](https://onlinelibrary.wiley.com/doi/10.1002/malq.201100081), Math Logic Quarterly, 58 (6):394-398 (2012), DOI:10.1002/malq.201100081, arXiv:[1106.4635](https://arxiv.org/abs/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. 

Meanwhile, Justin and I proved that RR is strictly intermediate between ZF and ZFC.