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BEBOP is a radial-velocity survey that monitors a sample of single-linedeclipsing binaries, in search of circumbinary planets by using high-resolutionspectrographs. Here, we describe and test the methods we use to identifyplanetary signals within the BEBOP data, and establish how we quantify oursensitivity to circumbinary planets by producing detection limits. This processis made easier and more robust by using a diffusive nested sampler. In theprocess of testing our methods, we notice that contrary to popular wisdom,assuming circular orbits in calculating detection limits for a radial velocitysurvey provides over-optimistic detection limits by up to $40\%$ insemi-amplitude with implications for all radial-velocity surveys. We performexample analyses using three BEBOP targets from our Southern HARPS survey. Wedemonstrate for the first time a repeated ability to reach a residual root meansquared scatter of $3~\rm m.s^{-1}$ (after removing the binary signal), andfind we are sensitive to circumbinary planets with masses down to that ofNeptune and Saturn, for orbital periods up to $1000~\rm days$.

BEBOP II: Sensitivity to sub-Saturn circumbinary planets using radial-velocities