  Shock interaction has been argued to play a role in powering a range of
optical transients, including supernovae (particularly the superluminous
class), classical novae, stellar mergers, tidal disruption events, and fast
blue optical transients. These same shocks can accelerate relativistic ions,
generating high-energy neutrino and gamma-ray emission via hadronic pion
production. The recent discovery of time-correlated optical and gamma-ray
emission in classical novae has revealed the important role of radiative shocks
in powering these events, enabling an unprecedented view of the properties of
ion acceleration, including its efficiency and energy spectrum, under similar
physical conditions to shocks in extragalactic transients. Here we introduce a
model for connecting the radiated optical fluence of non-relativistic
transients to their maximal neutrino and gamma-ray fluence. We apply this
technique to a wide range of extragalactic transient classes in order to place
limits on their contributions to the cosmological high-energy gamma-ray and
neutrino backgrounds. Based on a simple model for diffusive shock acceleration
at radiative shocks, calibrated to novae, we demonstrate that several of the
most luminous transients can accelerate protons up to energies $E_{\rm max}
\gtrsim 10^{16}$ eV, sufficient to contribute to the IceCube astrophysical
background. Furthermore, several of the considered sources$-$particularly
hydrogen-poor supernovae$-$may serve as "hidden" gamma-ray sources due to the
high gamma-ray opacity of their ejecta, evading constraints imposed by the
non-blazar Fermi-LAT background. However, adopting an ion acceleration
efficiency $\sim$ 0.3-1$\%$ motivated by nova observations, we find that
currently known classes of non-relativistic, potentially shock-powered
transients contribute at most a few percent of the total IceCube background.
