  The Baryon Acoustic Oscillations (BAO) from Integrated Neutral Gas
Observations (BINGO) radio telescope will use the neutral Hydrogen emission
line to map the Universe in the redshift range $0.127 \le z \le 0.449$, with
the main goal of probing BAO. In addition, the instrument optical design and
hardware configuration support the search for Fast Radio Bursts (FRBs). In this
work, we propose the use of a BINGO Interferometry System (BIS) including new
auxiliary, smaller, radio telescopes (hereafter \emph{outriggers}). The
interferometric approach makes it possible to pinpoint the FRB sources in the
sky. We present here the results of several BIS configurations combining BINGO
horns with and without mirrors ($4$ m, $5$ m, and $6$ m) and 5, 7, 9, or 10 for
single horns. We developed a new {\tt Python} package, the {\tt FRBlip}, which
generates synthetic FRB mock catalogs and computes, based on a telescope model,
the observed signal-to-noise ratio (S/N) that we used to compute numerically
the detection rates of the telescopes and how many interferometry pairs of
telescopes (\emph{baselines}) can observe an FRB. FRBs observed by more than
one baseline are the ones whose location can be determined. We thus evaluate
the performance of BIS regarding FRB localization. We found that BIS will be
able to localize 23 FRBs yearly with single horn outriggers in the best
configuration (using 10 outriggers of 6 m mirrors), with redshift $z \leq
0.96$; the full localization capability depends on the number and the type of
the outriggers. Wider beams are best to pinpoint FRB sources because potential
candidates will be observed by more baselines, while narrow beams look deep in
redshift. The BIS can be a powerful extension of the regular BINGO telescope,
dedicated to observe hundreds of FRBs during Phase 1. Many of them will be well
localized with a single horn + 6 m dish as outriggers.(Abridged)
