This paper reports on the $γ$-ray properties of the 2018 Galactic nova V392 Per, spanning photon energies $\sim$0.1 GeV to 100 TeV by combining observations from the Fermi Gamma-ray Space Telescope and the HAWC Observatory. In one of the most rapidly evolving $γ$-ray signals yet observed for a nova, GeV $γ$ rays with a power law spectrum with index $Γ= 2.0 \pm 0.1$ were detected over eight days following V392 Per's optical maximum. HAWC observations constrain the TeV $γ$-ray signal during this time and also before and after. We observe no statistically significant evidence of TeV $γ$-ray emission from V392 Per, but present flux limits. Tests of the extension of the Fermi/LAT spectrum to energies above 5 TeV are disfavored by 2 standard deviations (95\%) or more. We fit V392 Per's GeV $γ$ rays with hadronic acceleration models, incorporating optical observations, and compare the calculations with HAWC limits. 18 pages, 14 figures; revised for referee comments; section 5 revised