The requirement that a single event map to a single coordinate and vice versa is a rule for constructing a valid manifold. As such, it is not a property of the model, not of the laws of physics.This is your version of the "third postulate" of special relativity I was talking about. I agree that this postulate implies the kinematical character of boosts and ultimately the 4D geometry of space-time. However, my point is that one can imagine a world where this "consistency requirement" does not hold, but all other laws of physics are still valid.
There is no problem imagining coordinate systems which are not one to one and onto (just look at any road map with an inset). But I have great difficulty seeing how such could be a property of a hypothetical universe.