Cost-effective localization methods for Autonomous Underwater Vehicle (AUV) navigation are key for ocean monitoring and data collection at high resolution in time and space. Algorithmic solutions suitable for real-time processing that handle nonlinear measurement models and different forms of measurement uncertainty will accelerate the development of field-ready technology. This paper details a Bayesian estimation method for landmark-aided navigation using a Side-scan Sonar (SSS) sensor. The method bounds navigation filter error in the GPS-denied undersea environment and captures the highly nonlinear nature of slant range measurements while remaining computationally tractable. Combining a novel measurement model with the chosen statistical framework facilitates the efficient use of SSS data and, in the future, could be used in real time. The proposed filter has two primary steps: a prediction step using an unscented transform and an update step utilizing particles. The update step performs probabilistic association of sonar detections with known landmarks. We evaluate algorithm performance and tractability using synthetic data and real data collected field experiments. Field experiments were performed using two different marine robotic platforms with two different SSS and at two different sites. Finally, we discuss the computational requirements of the proposed method and how it extends to real-time applications.