3D Gaussian Splatting (3DGS) has recently emerged as a significant advancement in 3D scene reconstruction, offering real-time, high-resolution, and photorealistic rendering. Despite these advantages, 3DGS often compromises geometric accuracy for visual fidelity. In contrast, geometrically precise representations such as voxel grids, point clouds, and meshes are widely used in robotics, particularly with the increasing availability of high-accuracy LiDAR  and LiDAR-Inertial-Visual (LIV) systems. Recent research has utilized LiDAR priors to initialize 3D Gaussian s. However, the optimization processes in 3DGS can distort the original geometric information. Furthermore, existing methods mainly focus on offline 3DGS training, which cannot fully exploit the real-time capabilities of LIV systems. To address these limitations, we introduce MEGA, an edge-assisted online reconstruction approach with mesh-aligned 3DGS. MEGA facilitates online 3DGS training by leveraging incrementally available posed frames, colored LiDAR points, and triangle mesh faces from LIV systems. It employs a novel mesh-aligned representation to dynamically populate 3D Gaussian s based on triangle mesh faces. Additionally, it introduces an image-to-geometry alignment technique to resolve inconsistencies between frames and LiDAR priors. Extensive evaluations demonstrate that MEGA achieves superior rendering quality while preserving precise geometric information.