The dispersion measure—redshift relation of fast radio bursts (FRBs), DM (z ), has been proposed as a potential new probe of the cosmos, complementary to existing techniques. In practice, however, the effectiveness of this approach depends on a number of factors, including (but not limited to) the intrinsic scatter in the data caused by intervening matter inhomogeneities. Here, we simulate a number of catalogues of mock FRB observations, and use Markov Chain Monte Carlo techniques to forecast constraints, and assess which parameters will likely be best constrained. In all cases, we find that any potential improvement in cosmological constraints are limited by the current uncertainty on the diffuse gas fraction, fd(z ) . Instead, we find that the precision of current cosmological constraints allows one to constrain fd(z ) and possibly its redshift evolution. Combining Cosmic Microwave Background + Baryon Acoustic Oscillations + Supernovae + H0 constraints with just 100 FRBs (with redshifts), we find a typical constraint on the mean diffuse gas fraction of a few percent. A detection of this nature would alleviate the "missing baryon problem," and therefore highlights the value of localization and spectroscopic follow-up of future FRB detections.