Plasma-based accelerators are beginning to employ relativistic beams with unprecedented charge and ultrashort durations. These dense driver beams can drive wakes even in high-density plasmas ($\gtrsim {10}^{19}$ cm${}^{-3}$), where betatron radiation becomes increasingly important and begins to affect the dynamics of the accelerated beam. We show that betatron cooling leads to a strong structuring of the phase space of the beam. This gives rise to bunched, ring-like structures with positive radial position and momentum gradients, , population inversion of the amplitude of oscillation. We derive the characteristic timescales for this process analytically and confirm our predictions with multi-dimensional Particle-in-Cell simulations. The radiation-dominated regime of beam dynamics fundamentally alters the acceleration process and produces self-structured beams; Particle-in-Cell simulations of the resulting beams show that it becomes unstable to the ion-channel laser and lead to exponential amplification of radiation.