Microstructure reconstruction is an important and emerging aspect of computational materials engineering and multiscale modeling and simulation. Despite extensive research and fast progress in the field, the application of descriptor-based reconstruction remains limited by computational resources. Common methods for increasing the computational feasibility of descriptor-based microstructure reconstruction lie in approximating the microstructure by simple geometrical shapes and by utilizing differentiable descriptors to enable gradient-based optimization. The present work combines these two ideas for structures composed of non-overlapping ellipsoidal inclusions such as magnetorheological elastomers. This requires to express the descriptors as a function of the microstructure parametrization. Deriving these relations leads to analytical solutions that further speed up the reconstruction procedure. Based on these descriptors, microstructure reconstruction is formulated as a multi-stage optimization procedure. The developed algorithm is validated by means of different numerical experiments and advantages and limitations are discussed in detail.