I am unfamiliar with Shape Dynamics.
However, it appears to be another possible route toward a quantum theory of gravitation.
( see
https://en.wikipedia.org/wiki/Quantum_gravity#Candidate_theories )
soapbox:
Following the Correspondence Principle, one would expect each candidate theory to
reproduce much of the known experimental results
while making a new prediction that makes some headway
toward a successful formulation of quantum gravity.
Since quantum gravity has been stuck for a while (due to limited experimental guidance),
theorists try to make some progress.
Each candidate theory explores the implications of
holding sacred some declared set of fundamental concepts and principles,
while weakening or generalizing others.
Some are more popular, more well-funded, more respected, etc... than others.
Because of this, some ideas are more developed than others,
where "development" could mean progress toward success,
as well as progress in realizing that an approach may be wrong or too difficult
and that one must back up a few steps and try something else.
(My dissertation involved aspects of one of the Other Theories.)
Following your link to "Shape Dynamics: Relativity and Relationalism" by Flavio Mercati,
the chapter entitled "Relativity Without Relativity" (a title in the spirit of JA Wheeler)
https://oxford.universitypressschol...98789475.001.0001/oso-9780198789475-chapter-7
suggested to me to google it.
It led to
Relativity without relativity (2002)
Julian Barbour, Brendan Foster, Niall Ó Murchadha
https://arxiv.org/abs/gr-qc/0012089
The answer to your question may be implicit in there... but it's a rather technical presentation.
Another type of resource to seek out are dissertations.
Here's one I found by googling "Shape Dynamics"
https://www.imperial.ac.uk/media/im...tations/2020/Joshua-Hoffmann-Dissertation.pdf
The introduction of dissertations might be helpful.
The chapter title "Deriving Local Lorentz Invariance" caught my eye... but the chapter is rather involved.
One can also try to find seminars (say, on YouTube or on physics-department websites).
Here's one that seemed promising
Tim A. Koslowski (New Brunswick) gives a talk at the Mini-Workshop on the Foundations of Shape Dynamics (23 June, 2014) titled "Shape Dynamics". ( &t=19m46s looked promising... but it wasn't clear to me)
I have nothing more to offer
other than these strategies for finding answers to your questions.
Good luck.
