Accurate and robust localization is a fundamental requirement for service and inspection robots, particularly in feature-sparse indoor environments where traditional systems struggle due to a lack of distinct landmarks. While prior maps can enhance robustness, precise and compact maps capturing real-world details are often unavailable for new or frequently changing environments. This paper presents BIM-Loc, a novel discrepancy-aware LiDAR-based localization method that directly integrates Building Information Models (BIM) from the design phase. BIM-Loc simultaneously estimates trajectories aligned with the BIM coordinate system and identifies discrepancies between real-world observations and the as-designed BIM in an online fashion. Our core contributions include: (1) a novel multi-hit ray casting strategy for efficient BIM-point data association and projection of 3D observations into 2D texture space; (2) a pose graph optimization framework with BIM-integrated factors that enforces consistency among odometry, sequential scans, and BIM structures; and (3) a hierarchical Bayesian inference module that incrementally updates a continuous 2D surface representation for discrepancy detection, propagating updates from the pixel to the structure level. Extensive evaluations in both simulation and real-world applications demonstrate that BIM-Loc significantly outperforms state-of-the-art map-based methods in localization accuracy and robustness.