The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multi-grid solvers this approach allows to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol. ; Comment: Just accepted at Journal of Chemical Theory and Computation. This document is the unedited Authors version of a Submitted Work that was subsequently accepted for publication in the Journal of Chemical Theory and Computation, copyright American Chemical Society after peer review. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.jctc.6b00435