UK/USA passports: which to use when leaving NZ with transit in LAX to the UK as UK resident?

When leaving the UK, show your USA passport to the airline but have your UK passport if they do have an exit check (which are rare but are coming in, I am told.)
Upon arrival in the USA use your USA passport. When leaving the USA, you should have your USA passport on the paperwork but will need to show whichever passport you use to get into New Zealand, as the airline will need to be sure you have the needed visa or do not need visa.

If you have not decided yet, you should look into whether one or the other passport gives an easier access into NZ (which is most likely your USA passport.)

The next bit I had written because I misread your question, for the return journey.

For the exit from New Zealand use the same passport as you used to get into the country. For the airline, show both, so that they know you do not need a visa or esta for either country. When arriving in the USA you need to use your USA passport, even for just a transit.

When checking in and boarding the next flight, show both but have the USA one as the one on record. On arrival in the UK your UK passport will be the best.

When you leave the UK, you will not have to show a passport to UK authorities. The documents you'll need to show depend on whether you plan to enter New Zealand with your US passport or your UK passport. In some cases, it doesn't matter which passport you show.

• Check in in London with the passport that you plan to use in New Zealand. If that is your UK passport, explain that you do not need ESTA because you also have a US passport, and show it if asked.

• In transit in the US, show your US passport.

• On arrival in New Zealand, show whichever passport you've chosen to show in New Zealand.

• When you check in for the return journey, you can show either passport. As with the outbound journey, if you show your UK passport, you should also volunteer that you do not need ESTA because you also have a US passport.

• When you pass through New Zealand exit controls, show whatever passport you used when you entered. You want the government to connect your departure record with your arrival record.

• In transit in the US, show your US passport.

• On arrival in the UK, show your UK passport.

At any point, if anyone asks about your ability to remain in the UK when you've shown your US passport (or vice versa), you can simply tell them that you are a dual citizen, and show the other passport. Both countries allow dual citizenship.

If you'll be using your US passport in New Zealand, you could actually check in for every leg of the trip with the US passport. This would simplify things somewhat by allowing you to avoid questions about ESTA; you would show your UK passport only once, at the very end of your trip, to the UK Border Force when you reach the passport checkpoint on your return to the UK.

The reason for this is that the airline is not likely to check whether you have a right of abode in the UK. For example, US citizens are not required to have proof of onward travel when they fly to the UK. Therefore, for a flight to the UK, you can check in with either passport. Since this flight also involves transit in the US, you actual choices when checking in in New Zealand are to show either both passports or only your US passport.

The airline is not likely to check the conditions of your stay in the UK, but they might. If they do, tell them you are a citizen of the UK and show your passport. They probably will not ask why you chose to use your US passport, but if they do, explain that it is because you need to use it for the transit in the US.