This paper introduces BIMCaP, a novel method to integrate mobile 3D sparse LiDAR data and camera measurements with pre-existing building information models (BIMs), enhancing fast and accurate indoor mapping with affordable sensors. BIMCaP refines sensor poses by leveraging a 3D BIM and employing a bundle adjustment technique to align real-world measurements with the model. Experiments using real-world open-access data show that BIMCaP achieves superior accuracy, reducing translational error by over 4 cm compared to current state-of-the-art methods. This advancement enhances the accuracy and cost-effectiveness of 3D mapping methodologies like SLAM. BIMCaP's improvements benefit various fields, including construction site management and emergency response, by providing up-to-date, aligned digital maps for better decision-making and productivity. Link to the repository: https://github.com/MigVega/BIMCaP