Ouch!

I assume the tickets are on two separate PNRs. Unfortunately, you are pretty much at Air New Zealand's mercy in terms of their waiving fare increases, so they are the company you need to beg to help you. AA will give you token compensation for being late into LAX, and that is all.

Unfortunately, when you buy two separate tickets, you bear the risk of something like this happening, so have to plan in extra time at connections.

Good luck!

ETA: There is absolutely no cause to file a credit card dispute here, and it will only cause you more trouble. That is a terrible suggestion!