We study Mallows random permutations conditioned to avoid a given pattern $α$ of length~$3$. When the bias parameter is of the form $e^{β/n}$, we prove that these permutations converge to a non-trivial explicit deterministic permuton that depends on the pattern $α$ and on the parameter $β$. Along the way, we provide parametrizations for $α$-avoiding permutons, and establish a large deviation principle for uniform $α$-avoiding permutations. As a byproduct of the proof, we also obtain asymptotic estimates of two versions of $q$-Catalan numbers in the regime $q=e^{β/n}$.