This is begging the question though -- relativity presupposes the existence of a 4D spacetime upon which you can do your calculations, so at that moment when yo

This is begging the question though -- relativity presupposes the existence of a 4D spacetime upon which you can do your calculations, so at that moment when you state that the local reference frame is not accelerating, is the moment that you declared that the object is not rotating but the universe is. However relativity also admits the description of spacetime where the entire universe rotates and the object's edges are experiencing constant acceleration. One way to look at relativity is note that local observers can disagree about apparent distances, times, velocities, and accelerations, they are all able to use Lorentz transforms to answer the question of 'what would another observer see'. So in any relativistic description of this universe, all observers need to agree on whether the object in the middle is rotating or not -- but this appears to be a free parameter. In other words, General Relativity fails to fully follow Mach's principle on a global scale.