I've felt for a long time that Kohlberg's stages remain intact but take on a different character within Jewish Moral Development. For example, the notion of a covenantal relationship with HKB"H would correspond to level 5. 'Halicha Be-drachav', would correspond to stage 6, where abstract ethical principles guide behavior, and not concrete rules. Sifrei Chassidut often talk about a higher level, that of a Tzaddik, where one no longer perceives himself as an independent entity, rather as a pure manifestation of God's will.
Also - I don't believe that a higher stage means that someone is 'better', rather, 'more mature'. I will return to the notion of religious maturity (I've alluded to it in my post called "Gam Ki Yazkin"). A level 5 isn't necessarily 'frummer' than a level 3, but frumkeit at level 5 is more mature than frumkeit at level 3.
Kohlberg is a map. No reason to be dogmatic about his levels, but they're useful., even in a gestalt or constructivist approach to morality.