You're looking at the big time outfitters there mate, offering a full top notch 5+star hunting package. And to be honest you can only blame rich Arabs, Americans and Europeans for driving the price for such a package up so high. At the end of the day there are people willing to pay that price, those outfitters are in business and the two work hand in hand.
I know that Leithen will offer any type of package you wish/afford-but you need to talk to them.
https://www.leithenvalley.co.nz/ To be fair they are very nice people, are professional and conduct their hunting business to a very high standard. They established themselves as a hunting outfitter in NZ way back, everyone knocked them for doing so, but perseverance paid off and they now own a large chunk of the valley they live in, and can provide exceptional hunting opportunities. But as a hunter, you have to realize that the majority of their hunting packages are for people who are here to shoot a trophy animal in a short time frame, not experience the real way we kiwis hunt. Most kiwi hunters will hunt their whole life without even seeing an animal in the wild of the caliber clients are shooting on that place. I do know each year they bring some kids out from the US who for various reasons (funds,disability,etc) and circumstances may never get the opportunity, and take them hunting and fishing for a few days. Not sure how it works but I know they do it, so maybe worth an email to them?
Now if you are wanting a real deal adventure you're best to make some buddies on here, book some flights and come out. You'll get to hunt the real way, stay in a tent or hut, and see some neat country. There are some of us that could take you/guide you on private properties for a pretty fair daily fee, with a success fee to the landowner if you shoot an animal, as sometimes this is required to get onto the land we have access to. Guaranteed animals seen, and will put you in a position to take a shot on an animal. Will not be a huge 350+SCI red stag, but a real wild deer (or chamois, tahr, etc) in it's wild natural environment. Or if you are lucky you might be able to tag along for a couple of days with a local kiwi hunter for nothing, no guarantees on animals but a guarantee on a good holiday in an awesome country with amazing scenery and outdoors opportunities. And of course you will meet the odd good bastard or two.
With a bit of time here and research I would be surprised if you couldn't plan trip and come out here with your son and do a bit of hunting and a bit of fishing for less than $10,000-$15,000 flights/accommodation included.