Baryonic feedback is a leading contaminant in studying dark matter using cosmic shear. This has meant omitting much of the data during cosmological inference, or forward-modeling the spatial distribution of gas around dark matter halos using models for baryonic feedback, which introduces nuisance parameters and raises questions of potential model misspecification. We propose a novel method of "baryon nulling" using cross-correlations between shear maps and fast radio burst (FRB) dispersion measures. We use the FLAMINGO suite of hydrodynamical simulations, whose power spectra span a wide yet realistic range of feedback variations, to demonstrate that our nulling procedures reduces sensitivity to feedback at $k\approx 1$ Mpc${}^{-1}$ by a factor of a few, offering a valuable model-independent alternative to standard forward modeling procedures, which often assume a family of feedback models. Sensitive FRB surveys such as CHORD and the DSA-2000, and wide area surveys like Rubin, Roman, and Euclid, may enable the megaparsec-scale clustering of dark matter to be precisely constrained without a baryonic prior.