Multispectral and hyperspectral data fusion allows data restoration with increased spatial and spectral resolutions. A common approach, in particular with the presence of variant blur, is to solve an ill-posed inverse problem by minimizing a mixed criterion. This minimization usually requires an iterative gradient-based method. Still, this paper demonstrates the existence of a reachable explicit solution without the need to solve a Sylvester equation, allowing for a three order of magnitude timesaving factor wrt. iterative algorithms. We exploit this explicit resolution to accelerate the minimization by a Majorize-Minimize algorithm of a semi-quadratic convex edge-preserving criterion. We conduct experiments on realistic synthetic measurements for the James Webb Space Telescope and show that our proposed solution outperforms the state-of-the-art in computation time and quality of reconstruction.