Multiple imaging by a gravitational lens strongly correlates widely separated regions of the image, allowing reconstruction of both the lens potential and the unlensed source structure. Thus, lens inversion is analogous to self-calibration because both techniques construct a model of the distorting medium constrained by consistency requirements in the data. The lens inversion algorithm, LensClean, has assumed that the image reconstruction techniques used in radio astronomy do not introduce errors that affect the lens modeling. We extend LensClean to work directly from visibility data, with the option of adding self-calibration steps, and investigate the level of systematic errors present in reconstructions of a point source system, MG 0414 + 0534, and an extended emission system, MG 1654 + 1346. We find that CLEAN-reconstructed radio maps contain significant deconvolution errors that degrade both the accuracy of reconstructed lens images and broaden the uncertainty in physical parameters of the lens models. Lens model reconstructions, even with a less than perfect lens model, are generally a better fit to the visibility data than a standard CLEAN map. Rigorous analyses of radio observations should be performed directly on the visibility data.