Jech's The Axiom of Choice has a couple class permutation models, a review shows theorem 11.2, as well Problems 9.3, 9.4 which you may want to examine (they talk about Injection Principle and Surjection principles from classes to sets, and the failure of them)
Also, while not permutation per se, Monro constructs via symmetric class forcing a Dedekind-finite proper class (that is a class that every function into $\omega$ is bounded). You may find this proof in Independence results concerning Dedekind-finite sets (J. Austral. Math. Soc. 19 (1975), 35–46).