We build upon our recently developed microscopic theory for the tube confinement potential of rigid macromolecules to treat the relaxation of stress and orientation of fluids of topologically entangled needles during a continuous startup shear deformation. Two coupled and highly nonlinear evolution equations for stress and rod orientation are proposed. The novel feature is our ability to self-consistently relate the current stress-and orientation-dependent state of the fluid with an effective instantaneous relaxation time that combines (perturbed) reptation and transverse activated barrier hopping. The effective relaxation time follows from a microscopic, time-dependent prediction of the tube confinement potential during the deformation. Results for the strain and Weissenberg number dependence of the confinement potential are presented for two degrees of entanglement. As a natural consequence of the predicted stress dependence of the confinement potential, our self-consistent single-rod theory emergently manifests features normally associated with many-polymer convective constraint release, such as a monotonic flow curve and an effective relaxation time that (nearly) scales with the inverse of the deformation rate. Comparisons with the original Doi-Edwards theory reveal qualitative differences as a consequence of stress- and orientation-induced softening of the tube constraints. Severe tube dilation often occurs, and the complete destruction of transverse localization is possible depending on system-specific conditions.