Recent cosmological data and astrophysical observations, such as the Hubble tension and the increasing preference from galaxy surveys for dynamical dark energy, have begun to challenge the standard $\Lambda$-cold dark matter cosmological model. Primordial magnetic fields (PMFs) offer a mechanism to alleviate these tensions within the framework of the standard model. These fields source excess small-scale baryon clumping, which can speed up recombination and shrink the comoving sound horizon at the surface of last scattering. Computing the modified recombination history requires coupling the radiative transport of Lyman-$\alpha$ photons to compressible magnetohydrodynamic simulations. Since doing so is generically computationally intractable, we have developed a linearized treatment that self-consistently computes the modified recombination history in the presence of PMF-induced baryon clumping for fields with red-tilted spectra. The clumping factors we find are too small to alleviate outstanding cosmological tensions, but our general framework can be applied to other PMF spectra, and provides a significant theoretical step towards a complete account of recombination in the presence of small-scale baryon clumping. All data and analysis are publicly available at .