Despite promising astrometric signals, to date there has been no success in direct imaging of a hypothesized third member of the Sirius system. Using the Clio instrument and MagAO adaptive optics system on the Magellan Clay 6.5 m telescope, we have obtained extensive imagery of Sirius through a vector apodizing phase plate (vAPP) coronagraph in a narrowband filter at 3.9 microns. The vAPP coronagraph and MagAO allow us to be sensitive to planets much less massive than the limits set by previous non-detections. However, analysis of these data presents challenges due to the target's brightness and unique characteristics of the instrument. We present a comparison of dimensionality reduction techniques to construct background illumination maps for the whole detector using the areas of the detector that are not dominated by starlight. Additionally, we describe a procedure for sub-pixel alignment of vAPP data using a physical-optics-based model of the coronagraphic PSF.