  The Lyman-$\alpha$ forest is a valuable probe of dark matter models featuring
a scale-dependent suppression of the power spectrum as compared to
$\Lambda$CDM. In this work, we present a new estimator of the Lyman-$\alpha$
flux power spectrum that does not rely on hydrodynamical simulations. Our
framework is characterized by nuisance parameters that encapsulate the complex
physics of the intergalactic medium and sensitivity to highly non-linear
small-scale modes. After validating the approach based on high-resolution
hydrodynamical simulations for $\Lambda$CDM, we derive conservative constraints
on interacting dark matter models from BOSS Lyman-$\alpha$ data on large
scales, k<0.02(km/s)^(-1), with the relevant nuisance parameters left free in
the model fit. The estimator yields lower bounds on the mass of cannibal dark
matter, where freeze-out occurs through 3-to-2 annihilation, in the MeV range.
Furthermore, we find that models of dark matter interacting with dark
radiation, which have been argued to address the $H_0$ and $\sigma_8$ tensions,
are compatible with BOSS Lyman-$\alpha$ data.
