For a class $\mathcal{C}$ of permutations, we cannot expect to sort the permutations of $\mathcal{C}$ with less than $O(\log |\mathcal{C}_n|)$ comparisons, where by convention $\mathcal{C}_n := \mathcal{C} \cap S_n$.
In particular, when $\mathcal{C}$ is closed by subpatterns, it follows by Marcus-Tardos theorem (refined by J. Fox) that $|\mathcal{C}_n| \leq C^n$ where $C$ is the Stanley-Wilf constant of $\mathcal{C}$. This leads to the following question: is it possible to sort such a class using at most $O(n \log C)$ comparisons? This question is a strengthening of Question 1 in the paper 'Fast Sorting and Pattern-avoiding Permutations' by D. Arthur.
It seems possible to represent such a sorting strategy by a binary tree which would essentially mimic an 'unbalanced' merge-sort algorithm. Here is the idea: given a permutation $\pi$, we would look for a tree $T_{\pi}$ leaf-labeled by the points of $\pi$, such that for each node $u$ of $T_{\pi}$ the 'overlap' between the two child subtrees would be $O(\log C)$ (either worst-case or on average). However I suspect that a more involved structure is necessary to solve this problem; should it admit a positive solution.