Compensated isocurvature perturbations (CIPs) are opposite spatial fluctuations in the baryon and dark matter density. They can be generated for example in the curvaton model in the early Universe but are difficult to observe because their gravitational imprint nearly cancels. We therefore propose a new measurement method by searching for a spatial modulation of the baryon acoustic oscillation (BAO) scale that CIPs induce. We find that for a Euclid-like survey the sensitivity is marginally better than the WMAP cosmic microwave background (CMB) constraint, which exploits the CIP-induced modulation of the CMB sound horizon. For a cosmic-variance limited BAO survey using emission-line galaxies up to $zsim7$ the sensitivity is between stage 3 and stage 4 CMB experiments. These results include using CIP-galaxy cross-correlations, which improves the sensitivity by a factor of $sim2-3$ for correlated CIPs. The method could be further improved with an optimal estimator, similarly to the CMB, and could provide a useful cross-check of other CIP probes. Finally, if CIPs exist, they can bias cosmological measurements made assuming no CIPs. In particular, they can act as a super-sample fluctuation of the baryon density and bias measurements of the BAO scale. For modern BAO surveys, the largest 2$sigma$ CIP fluctuation allowed by Plancks 95% bound could bias BAO measurements of $H(z)$ by 2.2%, partially reducing the tension with the local $H_0$ measurements from 3.1$sigma$ to 2.3$sigma$.
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