This paper introduces RABBIT, a novel robot-assisted bed bathing system designed to address the growing need for assistive technologies in personal hygiene tasks. It combines multimodal perception and dual (software and hardware) compliance to perform safe and comfortable physical human-robot interaction. Using RGB and thermal imaging to segment dry, soapy, and wet skin regions accurately, RABBIT can effectively execute washing, rinsing, and drying tasks in line with expert caregiving practices. Our system includes custom-designed motion primitives inspired by human caregiving techniques, and a novel compliant end-effector called Scrubby, optimized for gentle and effective interactions. We conducted a user study with 12 participants, including one participant with severe mobility limitations, demonstrating the system's effectiveness and perceived comfort. Supplementary material and videos can be found on our website \hrefhttps://emprise.cs.cornell.edu/rabbit emprise.cs.cornell.edu/rabbit .