How does Kohlberg's moral reasoning work, can I measure it myself? Is there any way I can work out what stage Kohlberg would claim I was operating at? 

I do refer, when thinking about what is best, permissible, etc., to laws and other people's opinions, punishment etc., but not as conclusive judgments on what is right or wrong etc., only as one aspect of how to try to make that judgment myself. And while my actual judgments I would claim apply to anyone, not just myself, they are also provisional, and subject to revision, reappraisal. 

With some exceptions:

My happiness is often more valuable than other's, maybe just because I'm especially well set to work on that. Likewise I wouldn't be obliged to martyr myself for any reason, e.g. to save a million lives, though I would say the same of anyone. And certain actions are always and unequivocally wrong, for anyone, they are that heinous. 

I think enough and loosely enough about morality to say that I'm probably post conventional, and perhaps that's all that really does matter. But I wondered if I could learn more!
That can't be answered with one particular stage because:


  Another criticism of Kohlberg’s theory is that people frequently demonstrate significant inconsistency in their moral judgements.[26] This often occurs in moral dilemmas involving drinking and driving and business situations where participants have been shown to reason at a subpar stage, typically using more self-interest driven reasoning (i.e. stage two) than authority and social order obedience driven reasoning (i.e. stage four).[26][27] Kohlberg’s theory is generally considered to be incompatible with inconsistencies in moral reasoning. [...]  Krebs and Denton have also attempted to modify Kohlberg’s theory to account for a multitude of conflicting findings, but eventually concluded that the theory is not equipped to take into consideration how most individuals make moral decisions in their everyday lives.


Having said that Wikipedia also mentions that some tests have been developed based on Kohlberg’s theory anyway, e.g. DIT (presumably averaging reasoning over a number questions), so you could try to take some of those tests.

As far as I can tell from their description(s), the DIT tests only involve abstractly reasoning about some scenarios that don't involve sellf-interest of the test-taker, so they don't quite seem to address the criticism, and they probably won't capture your exceptions in that area either, e.g "My happiness is often more valuable than other's". Krebs and Denton found that self-serving bias affects most people when it comes to moral judgements:


  Self-serving biases. In contrast to the third-person moral judgments
  evoked by Kohlberg’s test, most real-life moral judgments are made in the first person and second person. As examples,
  people say such things as “I was wrong; I should never have done
  that” and “You should do your share.” Krebs and Laird (1998)
  found that first-person moral judgments made by participants
  about transgressions they had committed were significantly more
  lenient than the third-person judgments they made about transgressions
  committed by others, especially when others’ transgressions
  were committed against them