Multipath-based simultaneous localization and mapping (MP-SLAM) is a promising approach in wireless networks to jointly obtain position information of transmitters/receivers and information of the propagation environment. MP-SLAM models specular reflections at flat surfaces as virtual anchors (VAs), which are mirror images of base stations. Particle-based methods offer high flexibility and can approximate posterior probability density functions of the mobile agent state and the map feature states, (i.e., VA states) with complex shapes. However, they often require a large number of particles to counteract degeneracy in high-dimensional parameter spaces, leading to high computational complexity. Conversely using an insufficient number of particles leads to reduced estimation accuracy. In this paper, we introduce a low-complexity MP-SLAM algorithm using a sigma point (SP)-based implementation of the sum-product algorithm (SPA). We model the messages of continuous states of the agent and the VAs as Gaussian distributions and approximate nonlinearities via SP-transformations. This approach substantially reduces the computational complexity without decreasing accuracy. Since probabilistic data association yields Gaussian mixtures for the agent and VA states, we use moment matching to combine each mixture into a single Gaussian. Numerical results using synthetic and real data demonstrate that our method achieves significantly reduced computational runtimes compared to particle-based schemes, while exhibiting comparable (or even superior) localization and mapping performance.