The detection of the primordial $B$-mode spectrum of the polarized cosmic microwave background (CMB) signal may provide a probe of inflation. However, observation of such a faint signal requires excellent control of systematic errors. Interferometry proves to be a promising approach for overcoming such a challenge. In this paper we present a complete simulation pipeline of interferometric observations of CMB polarization, including systematic errors. We employ two different methods for obtaining the power spectra from mock data produced by simulated observations: the maximum likelihood method and the method of Gibbs sampling. We show that the results from both methods are consistent with each other, as well as, within a factor of 6, with analytical estimates. Several categories of systematic errors are considered: instrumental errors, consisting of antenna gain and antenna coupling errors, and beam errors, consisting of antenna pointing errors, beam cross-polarization and beam shape (and size) errors. In order to recover the tensor-to-scalar ratio, $r$, within a 10% tolerance level, which ensures the experiment is sensitive enough to detect the $B$-signal at $r=0.01$ in the multipole range $28 < ell < 384$, we find that, for a QUBIC-like experiment, Gaussian-distributed systematic errors must be controlled with precisions of $|g_{rms}| = 0.1$ for antenna gain, $|epsilon_{rms}| = 5 times 10^{-4}$ for antenna coupling, $delta_{rms} approx 0.7^circ$ for pointing, $zeta_{rms} approx 0.7^circ$ for beam shape, and $mu_{rms} = 5 times 10^{-4}$ for beam cross-polarization.
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