Is scale invariance axiomatic within physics, and if so, how does it get around the transition from the microscopic, quantum world, to the macroscopic, classical world?
Scale invariance is not present in most of the realistic physical theories and thus can in no way be considered axiomatic. It is already broken in some classical theories, but even those which possess scale invariance on the classical level (like Yang-Mills theory, for example) acquire a nontrivial scale dependence through renormalization when quantized. This is extremely important since it allows different interactions to dominate certain scales, which is exactly what we observe in the real world.
Only a limited number of models are actually scale-invariant on the quantum level. Those are called conformal field theories; they basically describe the critical points of the second-type phase transitions of other models.
Please note that scale invariance has nothing whatsoever to do with general coordinate transformations from General Relativity. The latter require a metric to transform alongside other fields thus describing the redundant degrees of freedom.