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The Baryon Acoustic Oscillations (BAO) from Integrated Neutral GasObservations (BINGO) radio telescope will use the neutral Hydrogen emissionline to map the Universe in the redshift range $0.127 \le z \le 0.449$, withthe main goal of probing BAO. In addition, the instrument optical design andhardware configuration support the search for Fast Radio Bursts (FRBs). In thiswork, we propose the use of a BINGO Interferometry System (BIS) including newauxiliary, smaller, radio telescopes (hereafter \emph{outriggers}). Theinterferometric approach makes it possible to pinpoint the FRB sources in thesky. We present here the results of several BIS configurations combining BINGOhorns with and without mirrors ($4$ m, $5$ m, and $6$ m) and 5, 7, 9, or 10 forsingle horns. We developed a new {\tt Python} package, the {\tt FRBlip}, whichgenerates synthetic FRB mock catalogs and computes, based on a telescope model,the observed signal-to-noise ratio (S/N) that we used to compute numericallythe detection rates of the telescopes and how many interferometry pairs oftelescopes (\emph{baselines}) can observe an FRB. FRBs observed by more thanone baseline are the ones whose location can be determined. We thus evaluatethe performance of BIS regarding FRB localization. We found that BIS will beable to localize 23 FRBs yearly with single horn outriggers in the bestconfiguration (using 10 outriggers of 6 m mirrors), with redshift $z \leq0.96$; the full localization capability depends on the number and the type ofthe outriggers. Wider beams are best to pinpoint FRB sources because potentialcandidates will be observed by more baselines, while narrow beams look deep inredshift. 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 welllocalized with a single horn + 6 m dish as outriggers.(Abridged)

The BINGO Project IX: Search for Fast Radio Bursts -- A Forecast for the BINGO Interferometry System