This paper establishes that the Field-Wave sector of the locked SU(2) Skyrme model, where an emergent Maxwell field is constructed from two pion components, appears as the leading asymptotic sector of the full nonlinear dynamics in the small-amplitude regime. Working in the massless, small-field limit, we prove two main results: (1) For initial data of size O(ε) in Sobolev norms, solutions of the full Skyrme system are approximated by ε times the linear wave solution with error O(ε³) on time intervals independent of ε. (2) For Field-Wave initial data, the emergent composite field strength F_μν(U) satisfies Maxwell's equations up to error O(ε³), with approximately conserved Maxwell stress-energy tensor. These results show that the Field-Wave/Maxwell sector identified in Paper II is not just a special class of configurations, but emerges naturally as the leading term in a controlled asymptotic expansion of the full nonlinear Skyrme dynamics. The analysis uses quasilinear hyperbolic PDE theory, energy estimates, and systematic small-amplitude expansions with explicit error bounds. This work is part of a program to realize both matter (Skyrmions) and radiation (Maxwell waves) as different configurations of a single continuous SU(2) Skyrme field.