The co-evolution of stars and supermassive black holes over cosmic time has shaped the history of the Universe, and understanding their relative balance is key to understanding galaxy evolution. Luminous infrared galaxies in the local Universe are an ideal population to study this relationship thanks to their high star formation rates and a range of emission from active galactic nuclei (AGN, quantified as the AGN fraction). We use Spitzer/IRS spectra of Luminous Infrared Galaxies in the GOALS sample to study mid-infrared atomic fine-structure lines tracing star formation and black hole accretion. Since many important lines are faint and undetected in the majority of the sample, we constrain the balance of star formation and black hole accretion across GOALS by stacking Spitzer/IRS spectra, revealing the relationship between 12.8 micron [NeII], 14.3 micron [NeV], 15.6 micron [NeIII], and 25.9 micron [OIV] luminosity as a function of AGN fraction. This allows us to measure the balance of star formation and black hole accretion across the GOALS sample. Furthermore, we use these results to constrain the contribution of star formation to tracers of black hole accretion in order to improve the accuracy of black hole accretion rate calculations. Luminous infrared galaxies are more common at cosmic noon, and understanding their conditions in the local Universe will provide an important reference for understanding the evolution of galaxies over cosmic time. Our study, which probes lines below the noise in Spitzer data provides the groundwork for future JWST observations with the MIRI instrument.