Thanks! I have managed to route something similar and yes the random AA segment is indeed a bit odd (it also shows up when flying via ORD instead of SFO, however, depending on the date it is possible to route with UA instead of AA. I'm wondering if it is possible to get the 5k $ fare when routing LX and LH in First, so far I've only been able to find prices of 7k+ $, 5k+ $ when only routing one of the flights in First.
Essentially my hard dates are conference in NYC on the 6th and 7th of June and conference in TYO on the 10th and 11th of July. So I'd like to fly from/back to Europe as close to before and after the two conferences as possible, while the NYC-AKL and AKL-TYO legs are totally flexible. But I would like a week or two in TYO as well as in AKL.

Here is what I found. I might just end up trying to upgrade with miles on the LH,LX legs. (Which you can luckily do with M&M despite only being booked in P):



EDIT: Got it Sub 3000.- CHF: