  We explore a novel method for measuring the average temperature of the
$\nu_x$ component in Type-II core-collapse supernovae. By measuring neutral
current incoherent neutrino-Argon interactions in DUNE we can obtain spectral
information for the combination of all active neutrino species. Combining this
all-neutrino spectral information with detailed charged current measurements of
the electron neutrino and electron anti-neutrino fluxes from DUNE and
Hyper-Kamiokande, we can infer the average temperature for the remaining
neutrino species in the $\nu_x$ component to within a factor two for most cases
and to 30% for a small range of average $\nu_x$ temperatures. Due to the
limited energy range of the emitted photons from incoherent neutral current
interactions on Argon, the $\nu_x$ temperature reconstruction demonstrates a
degeneracy in the one and two sigma credible regions. Furthermore, while large
uncertainties on the NC cross-section penalize this measurement, we examined
the efficacy of constraining NC cross-section uncertainties on improving
$\nu_x$ measurements. We found that if additional measurements of
B(M1$\uparrow$) 1$^+$ excited state transitions in Argon are able to reduce
correlated cross section uncertainties from 15% to 7%, the size of the
$1\sigma$ allowed regions for $T_{\nu_x}$ becomes sample size limited, and
approaches the case where there are no uncertainties on the cross-section.
