Laboratory automation requires reliable and precise handling of microplates, but existing robotic systems often struggle to achieve this, particularly when navigating around the dynamic and variable nature of laboratory environments. This work introduces a novel method integrating simultaneous localization and mapping (SLAM), computer vision, and tactile feedback for the precise and autonomous placement of microplates. Implemented on a bi-manual mobile robot, the method achieves fine-positioning accuracies of +/- 1.2 mm and +/- 0.4 degrees. The approach was validated through experiments using both mockup and real laboratory instruments, demonstrating at least a 95% success rate across varied conditions and robust performance in a multi-stage protocol. Compared to existing methods, our framework effectively generalizes to different instruments without compromising efficiency. These findings highlight the potential for enhanced robotic manipulation in laboratory automation, paving the way for more reliable and reproducible experimental workflows.