  (Abridged)
  Context: In the previous paper in this series, we identified that a
pentagonal arrangement of five telescopes, using a kernel-nulling beam
combiner, shows notable advantages for some important performance metrics for a
space-based mid-infrared nulling interferometer over several other considered
configurations for the detection of Earth-like exoplanets around solar-type
stars.
  Aims: We aim to produce a physical implementation of a kernel-nulling beam
combiner for such a configuration, as well as a discussion of systematic and
stochastic errors associated with the instrument.
  Methods: We developed a mathematical framework around a nulling beam
combiner, and then used it along with a space interferometry simulator to
identify the effects of systematic uncertainties.
  Results: We find that errors in the beam combiner optics, systematic phase
errors and the RMS fringe tracking errors result in instrument limited
performance at $\sim$4-7 $\mu$m, and zodiacal limited at $\gtrsim$10 $\mu$m.
Assuming a beam splitter reflectance error of $|\Delta R| = 5\%$ and phase
shift error of $\Delta\phi = 3$ degrees, we find that the fringe tracking RMS
should be kept to less than 3 nm in order to be photon limited, and the
systematic piston error be less than 0.5 nm to be appropriately sensitive to
planets with a contrast of 1$\times 10^{-7}$ over a 4-19 $\mu$m bandpass. We
also identify that the beam combiner design, with the inclusion of a well
positioned shutter, provides an ability to produce robust kernel observables
even if one or two collecting telescopes were to fail. The resulting four
telescope combiner, when put into an X-array formation, results in a
transmission map with a relative signal-to-noise ratio equivalent to 80% of the
fully functioning X-array combiner.
