  Recent cosmological tensions have rekindled the search for models beyond
$\Lambda$CDM that cause a suppression of the matter power spectrum. Due to the
small scales accessible to Lyman-$\alpha$ data they are an excellent additional
tool to probe such models. In this work we extend a recently-developed approach
for using Lyman-$\alpha$ data to constrain the power spectrum suppression
caused by almost any mixture of cold and non-standard dark matter. We highlight
the steps involved in the development of a corresponding likelihood that will
be publicly released upon publication of this work. We study three examples of
models suppressing the power spectrum, namely feebly interacting dark matter,
dark matter interacting with baryons, and mixed cold+warm dark matter. The
latter two can be well constrained from Lyman-$\alpha$ data, and we derive
novel conclusions on the cosmologically allowed parameter spaces, including
finding a mild preference for non-zero interactions between dark matter and
baryons. The consistency of the constraints obtained on these models highlight
the robustness and flexibility of the likelihood developed here.
