With all due respect to the people living in Hamilton, but what are you going to do with the remaining 3.5 days once you covered Hamilton and surroundings?
I think there are probably better places to see if one has only a limited time to travel, from a touristy point of view.
Hamilton - Rotorua = 200 KM (to/fro)
Hamilton - Matamata = 150 KM(to/fro)
Hamilton - Waitamo = 150 KM(to/fro)
Hamilton - Taupo = 400 KM(to/fro)
4 days/3 nights should be sufficient enough for us to cover these places. You have bear in mind that we are travelling with three kids between the ages of 1-3 and you know how kids could be cranky during road travel. We may have to stop multiple times during the drive to let them out for few mins of fresh air else they would be difficult to handle.
We are just covering main attractions as we are travelling as a group.We will have ample time to cover NZ we move to permanently next year if our SMC visa comes through.
I believe that your planning is not well-thought out and requires a lot of unnecessary back-and-forth driving for no other reason except simply because you believe Hamilton makes a wonderful base for travelling around the area.
Many people (residents) and tourists choose to drive straight through Hamilton (some like me don't want to stop at Hamilton at all, preferring to make my stop in Cambridge where it is much lovelier. My kid loves the park and we like to buy seasonal fruits from Cambridge markets and spend a hour relaxing, letting her run around, feed the ducks etc.
Anyway if you are becoming a resident soon, it may not matter in the long run how you plan the trip. But if you want to maximize time and perhaps enjoy the trip more, you could make Rotorua your base instead.
Spend only 1 night in Hamilton after Matamata. The next day drive to Waitomo and spend the afternoon in Rotorua. You have a full day in Rotorua on the 3rd day and can visit Taupo on the 4th. Drive back to Auckland on your last day. Rotorua and Taupo are more scenic tourist towns with several attractions that appeal to families.
Do not mistake NZ distances for what you are used to driving. The roads here are windier, the speed limits are lower, and you go through every small town on the way (as in right through the middle of town). What you think will take an hour will take 2 hours. I love Taupo, but I would never consider driving there for a day trip, and I'm only 140k away.