  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 over-optimistic detection limits by up to $40\%$ 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 we are sensitive to circumbinary planets with masses down to that of
Neptune and Saturn, for orbital periods up to $1000~\rm days$.
