Bearings-only target tracking using raw acoustic data typically achieves better tracking performance at lower signal-to-noise ratios (SNR) compared to approaches that rely on direction-of-arrival (DOA) preprocessing. However, using raw data directly as observations renders standard Kalman filter-type algorithms inapplicable, since the likelihood when using raw acoustic data depends only on second-order statistics. While particle filter-based methods can overcome this limitation, they are often computationally demanding, particularly in high-dimensional state spaces. To address this, we present a maximum a posteriori-based method to approximate the Bayesian filter recursions, which can handle likelihoods that depend only on the second-order statistics of the observations. The method formulates the observation update as an optimization problem and applies a Laplace approximation to estimate the posterior mean and covariance. This enables the direct use of raw observations while avoiding the computational cost associated with particle filtering. The method is validated through simulations and real-world data from a sea trial. Results show that the proposed approach achieves robust tracking performance at low SNR, outperforming the traditional target-tracking approach that uses DOA-based preprocessing of the acoustic data. Hence, it offers a computationally efficient alternative to particle filters for target tracking applications that utilize raw acoustic data.