We compute transverse and longitudinal diffractive structure functions to full next-to-leading order accuracy in the dipole picture of deep inelastic scattering. Our calculation uses the standard light-cone perturbation theory method for the partonic content of the virtual photon, together with the Color Glass Condensate description of the target color field. Our result includes as a subset the contribution calculated earlier. We show that there is a rapidity divergence that can be factorized into the BK evolution of the target Wilson lines, and that all other divergences cancel. We emphasize the robustness of a fixed invariant mass as the definition of the final state, leading to a specific pattern of divergence cancellation between real and virtual contributions. Unlike for jets or identified hadrons, there is no fragmentation function or jet definition to absorb collinear divergences for this inclusive observable.