This is only an answer to the request for references about the expansion.

[Theorem 7.4][1] in the book *The Ambient Metric* proves that any hyperbolic metric on a conformally compact manifold has an expansion of the type you specify.
Note that in the three-dimensional case you are asking about, $h_4$ is determined by $h_2$, but $h_2$ isn’t determined solely by the conformal boundary.
I’m not sure how to interpret this in the context of hyperbolic geometry.


  [1]: https://arxiv.org/pdf/0710.0919.pdf#page66