Match the permutations!
JavaScript, 64 57 bytes
4 bytes removed thanks to Martin Ender.
^(?!.*([^])(.*\1){3}]?)[$$[-^?!!.'''-*11{33}5577\\-]{57}$

Try it here.
Explanations (outdated)
^                                  # Beginning of the string.
(?!.*                              # Match only the strings that don't contain...
  (.)(.*\1){4}                     #     5 occurrences of the same character.
  [^1]?[^1]?                       #     Something that doesn't matter.
)
[]zzz^(?!!!.**[)\\1{{44}}666]{64}  # 64 occurrences of these 16 characters.
                                   # Some are duplicated to make sure the regex
                                   # contains 4 occurrences of each character.
\z                                 # End of the string.