In the last decade, novel hyperspectral cameras have been developed with particularly desirable characteristics of compactness and short acquisition time, retaining their potential to obtain spectral/spatial resolution competitive with respect to traditional cameras. However, a computational effort is required to recover an interpretable data cube. In this work we focus our attention on imaging spectrometers based on interferometry, for which the raw acquisition is an image whose spectral component is expressed as an interferogram. Previous works have focused on the inversion of such acquisition on a pixel-by-pixel basis within a Bayesian framework, leaving behind critical information on the spatial structure of the image data cube. In this work, we address this problem by integrating a spatial regularization for image reconstruction, showing that the combination of spectral and spatial regularizers leads to enhanced performances with respect to the pixelwise case. We compare our results with Plug-and-Play techniques, as its strategy to inject a set of denoisers from the literature can be implemented seamlessly with our physics-based formulation of the optimization problem.