BINGO is a project aiming to set the grounds for large-scale bolometric neutrinoless double-beta-decay experiments capable of investigating the effective Majorana neutrino mass at a few meV level. It focuses on developing innovative technologies to achieve a very low background index, of the order of 10−5 counts/(keV kg yr) in the region of interest. The BINGO demonstrator, called MINI-BINGO, will be composed of Li2MoO4 and TeO2 crystals coupled to bolometric light detectors designed to investigate the promising double-beta-decay isotopes 100Mo and 130Te. This will allow us to reject a significant background in bolometers caused by surface contamination from α-active radionuclides by means of light yield selection. In addition, BINGO introduces new methods to mitigate other sources of background, such as surface radioactive contamination, external γ radioactivity, and pile-up due to random coincidence of background events. This paper focuses on the description of an innovative assembly designed to reduce the passive materials in line of sight of the detectors, which is expected to be a dominant source of background in next-generation bolometric experiments. We present the performance of two prototype modules -- housing four Li2MoO4 crystals in total -- operated in the Canfranc underground laboratory in Spain within a facility developed for the CROSS double-beta-decay experiment.