How does one go about rigorously deriving special relativistic dynamics (both relativistic mechanics and electrodynamics) from quantum electrodynamics? Is this even possible from the mathematical structure of QED?
In a comment you clarify
but of course reproducing Maxwell's Equations as well"
The way mainstream physics sees this is that the classical level is emergent from the quantum level. To desire the exact equations for classical to come from the equations of quantum is like asking for the exact equations of thermodynamics to come out of statistical mechanics.
In this blog article by Motl, "How classical fields, particles emerge from quantum theory " the method is described, how to go from QED to classical light.
It is the data and the observables predicted that have to be consistent, not the mathematics, imo.