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Recent cosmological tensions have rekindled the search for models beyond $Λ$CDM that cause a suppression of the matter power spectrum. Due to the small scales accessible to Lyman-$α$ data they are an excellent additional tool to probe such models. In this work we extend a recently-developed approach for using Lyman-$α$ 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-$α$ 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.