There is some evidence that Soft Gamma-Ray Repeaters (SGR) may be neutron stars undergoing structural adjustments that produce the observed transient $\gamma$-ray events. We here consider the physics of Alfv\'en wave propagation and dissipation in the closed field-line region of the disturbed magnetosphere surrounding such an object, and show that the charged particle gas is unbounded along those field lines with the largest radius of curvature, leading to a highly dynamic (locally super-Eddington) outflow. Within this medium, the synchrotron emission is strongly self-absorbed and the plasma is optically thick to scattering. The radiation distribution in the fluid rest frame therefore approximates that of a blackbody in LTE with the expanding gas. However, the observed spectrum is modified by several factors, including (1) the frequency-dependent location of the photosphere and (2) the angle-dependent boosting for emitting elements at different inclination angles relative to the observer. We show that the spectrum calculated in this way is a reasonable fit to the observed SGR burst spectra, with a characteristic temperature correlated to the burst luminosity. The inferred source distance to SGR 0526-66 and SGR 1806-20 is consistent with their apparent positioning within the LMC (the former) and the Galactic plane (the latter).