Plasmonic nanopatch antennas that incorporate dielectric gaps hundreds of picometers to several nanometers thick have drawn increasing attention over the past decade because they confine electromagnetic fields to grossly sub-diffraction limited volumes. Substantial control over the optical properties of excitons and color centers confined within these plasmonic cavities has already been demonstrated with far-field optical spectroscopies, but near-field optical spectroscopies are essential to an improved understanding of the plasmon-emitter interaction at the nanoscale. Here, we characterize the intensity and phase-resolved plasmonic response of isolated nanopatch antennas with cathodoluminescence microscopy. Further, we explore the distinction between optical and electron-beam spectroscopies of coupled plasmon-exciton heterostructures to identify constraints and opportunities for future nanoscale characterization and control of hybrid nanophotonic structures. While we observe substantial Purcell enhancement in time-resolved photoluminescence spectroscopies, negligible Purcell enhancement is observed in cathodoluminescence spectroscopies of hybrid nanophotonic structures. The substantial differences in measured Purcell enhancement for electron-beam and laser excitation can be understood as a result of the different selection rules for these complementary experiments. These results provide a fundamentally new understanding of near-field plasmon-exciton interactions in nanopatch antennas that is essential to myriad emerging quantum photonic devices.