ABSTRACT BEBOP is a radial-velocity survey that monitors a sample of single-lined eclipsing binaries, in search of circumbinary planets by using high-resolution spectrographs. Here, we describe and test the methods we use to identify planetary signals within the BEBOP data and establish how we quantify our sensitivity to circumbinary planets by producing detection limits. This process is made easier and more robust by using a diffusive nested sampler. In the process of testing our methods, we notice that contrary to popular wisdom, assuming circular orbits in calculating detection limits for a radial-velocity survey provides overoptimistic detection limits by up to $40{{\ \rm per\ cent}}$ in semi-amplitude with implications for all radial-velocity surveys. We perform example analyses using three BEBOP targets from our Southern HARPS survey. We demonstrate for the first time a repeated ability to reach a residual root mean squared scatter of $3\, \rm m\, s^{-1}$ (after removing the binary signal), and find that we are sensitive to circumbinary planets with masses down to that of Neptune and Saturn, for orbital periods up to $1000\, \rm d$.