Hi
I agree that you're probably best avoiding LAX mum flew thru them when she visited and said it was a nightmare.
If you cant change your flight and are set on flying to AKL then I would probably opt for the Campervan trip and the journey between the two cities is awesome and shows some of the real New Zealand lifestyle.
Although I think there is a charge if you are not returning your campervan to the place you picked it up. You may want to look into relocating a campervan that was left in AKL but belongs to a Wellie company? If you do that you usually get a good deal but do have to put down a $1000 bond incase of any damages (or something like that).
The flight to Wellie would of course be quicker just watch the winds in Wellie when you land
There are also lots of cheap flights going on Air NZ grabaseat and of course pacific blue now.
If you havent booked your flights why not look into Emerates as their flights are getting slightly cheaper and are accommodating more direct routes.
I think one thing to remember is the jet lag when you arrive. It usually takes me a few days to recover so you'll have to take that into consideration especially with your kids.
Good luck with it all !