  The current experiments searching for neutrinoless double-$\beta$
($0\nu\beta\beta$) decay also collect large statistics of Standard Model
allowed two-neutrino double-$\beta$ ($2\nu\beta\beta$) decay events. These can
be used to search for Beyond Standard Model (BSM) physics via $2\nu\beta\beta$
decay spectral distortions. $^{100}$Mo has a natural advantage due to its
relatively short half-life, allowing higher $2\nu\beta\beta$ decay statistics
at equal exposures compared to the other isotopes. We demonstrate the potential
of the dual read-out bolometric technique exploiting a $^{100}$Mo exposure of
1.47 kg $\times$ y, acquired in the CUPID-Mo experiment at the Modane
underground laboratory (France). We set limits on $0\nu\beta\beta$ decays with
the emission of one or more Majorons, on $2\nu\beta\beta$ decay with Lorentz
violation, and $2\nu\beta\beta$ decay with a sterile neutrino emission. In this
analysis, we investigate the systematic uncertainty induced by modeling the
$2\nu\beta\beta$ decay spectral shape parameterized through an improved model,
an effect never considered before. This work motivates searches for BSM
processes in the upcoming CUPID experiment, which will collect the largest
amount of $2\nu\beta\beta$ decay events among the next-generation experiments.
