The properties of the permutation are dependent upon the encryption key, so a client being able to select malicious inputs to get long cycles implies either tha

The properties of the permutation are dependent upon the encryption key, so a client being able to select malicious inputs to get long cycles implies either that the client knows the AES key, or that the client has broken AES. In any case, as I mentioned in a sibling comment, with 3 AES encryptions one can construct a 122-bit balanced Feistel cipher with a constant amount of work.