The construction industry is one of the most significant consumers of environmental resources worldwide. Faced with the need to produce new buildings, but without further burdening the environment, attempts to improve social, economic, and environmental indicators have turned attention to building construction in recent decades. The objective of this research is to develop a novel framework to assess the most sustainable choice of materials applied to the construction of low-income buildings, according to the three pillars of the Triple Bottom Line (TBL). A BIM-LCSA-FAHP-based model was proposed with the creation of nine different scenarios, where the materials of the structure (precast concrete, cast-in-place concrete, and structural masonry), painting (PVA water-based and acrylic), and roofing (ceramic and fiber cement tiles) varied. The proposed procedure consists of the elaboration of a 3D Building Information Modeling (BIM) model, for which the parameters described above were evaluated according to the Life Cycle Sustainability Assessment (LCSA)-TBL-based criteria, divided into ten sub-criteria, that includes: (1) environmental (acidification, eutrophication, global warming, ozone depletion, smog formation, primary energy, non-renewable energy, and mass total), (2) economic (construction cost) and (3) socio-political issues (community impact). Finally, the Fuzzy Analytical Hierarchy Process (AHP) was used as a multi-criteria decision-making technique that helps in aggregating and classifying the impacts of each scenario in a sustainability index (SI). Regarding the best option for low-income construction, the results indicated that precast concrete when combined with acrylic paint and fiber cement tiles (scenario 3) proved to be the most advantageous and achieved first place in the sustainability index (SI) developed in this work. This methodology is replicable for different construction typologies and several categories of materials, making it a robust decision-aiding tool for engineers, architects, and decision makers.