Detectability of the 21 cm signal with BINGO through cross-correlation with photometric surveys

21 cm intensity mapping (HI IM) can efficiently map large cosmic volumes with good redshift resolution, but systematics and foreground contamination pose major challenges for extracting accurate cosmological information. Cross-correlation with galaxy surveys offers an efficient mitigation strategy, as both datasets have largely uncorrelated systematics. We evaluate the detectability of the 21 cm signal from the BINGO radio telescope by cross-correlating with the LSST photometric survey, given their strong overlap in area and redshift. Using lognormal simulations, we model the cosmological signal in the BINGO frequency range (980 - 1260 MHz), incorporating thermal noise, foregrounds, and cleanup. The LSST simulations include uncertainties in photometric redshift (photo-z) and galaxy number density in the first three redshift intervals (mean redshift approximately equal to 0.25, 0.35, 0.45), corresponding to the expected performance after 10 years of the survey. We show that photo-z errors significantly increase the noise in the cross-correlation, reducing its statistical significance to levels comparable to those of the autocorrelation. Still, the HI signal remains detectable through the cross-correlation, even with photo-z uncertainties similar to those of the LSST. Our results corroborate the feasibility of this approach under realistic conditions and motivate further refinement of the current analysis methods.