I would like to clarify if the Minkowski subspace of points inside the null hypersurface (light-cones) at each point's tangent space is just a homogeneous subspace(transitive under G) for the Poincare group G(meaning the semidirect product of translations and the proper orthochronous Lorentz group) or a principal homogeneous subspace(G-torsor), i.e. simply transitive under G.
I would tend to think that the physical notion of the special relativistic principle with a constant c at each frame means velocities of massive particles can be boosted from any point inside each of the future light cones to any other point's lightcone preserving the space, and this would imply that the action of the Poincare group on the bundle of future timelike directions is not just transitive but simply transitive, is this assumption flawed?
Your comments seem to justify transitivity, but not the other half of simple transitivity, which is freedom/uniqueness. Given any distinct $v,w$ in the future light cone, there will be many different Lorentz transformations sending $v$ to $w$: if you let $\phi$ be a boost such that $\phi(v)=w$, then we also have $R\circ\phi(v)=w$ for any spatial rotation $R$ about $w$.
Indeed, it's clear just from dimension counting that (a subspace of) Minkowski space can't be a torsor for such a big group.