How about this scenario: picture yourself orbiting a cylindrical mirror such that a light pulse emitted from the orbiter returns on the same path and the orbiter receives the returning light pulse at the same point, having done one orbit. Would there be an anomalous result in the measured distances, having calculated the radius (time taken on light path/2) vs circumference measured by the orbiter? The point of this is that when you calculate the circumference, working from the light signal measurement, it doesn't conform with the circumference measured by the orbiter i.e it will be contracted, so if the radius was calculated from the orbiter it won't match with the measured radius. Hope that makes sense. Which Idoubt it will. It's nearly frying my brain.
Edit. Just thought, it can't be a light signal that's released but a massive body (another observer) that is released and returns at the same time as the orbiter.