Metallic nanoparticle-on-a-mirror (NPoM) cavities enable extreme field confinement in sub-nm gaps, leading to unrivaled performance for nonlinear processes such as surface-enhanced Raman scattering (SERS). So far, prevailing experimental approaches based on NPoMs have been performed by means of free-space light excitation and collection under oblique incidence, since the fundamental radiatively-coupled NPoM mode does not scatter in the normal direction. Retaining this working principle, here we numerically show that plasmonic cavity modes in NPoM configurations can be efficiently excited in an integrated SERS approach through TM guided modes of silicon nitride (SiN) waveguides. Intensity enhancements beyond 10$^{5}$ can be achieved for gap spacings around 1 nm. So as to reduce unwanted SiN Raman background, the output Stokes signals are transferred to transversely placed waveguides, reaching coupling efficiencies of up to 10%. Geometrical parameters such as the gap thickness as well as the radius and position of the nanoparticle provide full control over the main spectral features, thereby enabling us to engineer and drive the optical response of NPoMs for high-performance SERS in Si-based photonic integrated platforms.