This is the kicker... Were the DFW-LAX and LAX-AKL tickets bought together or separately? If together then AA should have reaccommodated you somehow. If separate, then you sort of owned that risk unfortunately. As another poster pointed out, it may have turned out much better had you called NZ as soon as the mech looked likely and asked them for help. I don't know how they roll, but it couldn't have gone much worse.

Regarding the dispute, if they were bought together as one itinerary I would understand disputing the original charge as you paid for travel to New Zealand and didn't get what you paid for. I wouldn't feel comfortable disputing the other charge however as it's one that you understood and agreed to at the time, and the services were indeed provided. Of course you could discuss it with the card issuer and see what they think.