I'm not sure if what you write makes sense. Take one simple example: a flat Minkowski spacetime, empty except for a few light particles (so that their influence on the metric is negligible). This means that special relativity applies, and it's clearly consistent with GR.

Accelerated motions are not going to be relative in this universe, just like they aren't in Newton's theory. You can of course observe an accelerating particle and insist on using coordinates in which it remains in the origin (which is sometimes useful, as in e.g. the Rindler coordinates), but in this coordinate system, the universe will not have the above listed properties in any meaningful sense.