Welcome to the world of carry on luggage. This is a hard part of traveling. The more experienced you get the better you are at carry on luggage. I fall into one of two categories with my laptop. Most planes allow you a carry on plus a laptop. There is no problem with those flights. One carry on restrictions on some airlines means you have to be a bit flexible. Air New Zealand a couple of years ago stuck to their one carry on policy. Also the flight down here can go either way. .
What I’ve learned to do is leave enough room in my backpack for the laptop minus all the goodies. I don’t trust checking it through with the rest of the baggage and would rather have it on my person. If the airline says no then it goes in the backpack and the rest of the stuff goes checked.
Mind you that hasn’t happened for a few years now but I still am prepared.
A couple of other notes on carry ons. If you’ve ever had an airline lose your bags, you will learn very quick to carry a change of clothes. Expect that when you take the flight to McMurdo the weather will turn bad and the plane will be forced to return to Christchurch. All your luggage remains on the plane and you will have to live out of your carry on. It’s not uncommon for flights to be backed up for several days.They make you wear the issue clothing and boots on the plane every now and then someone packs their shoes and the flight boomerangs leaving them without anything to wear on their feet.
The flight down gives you a very limited carry on allowance. It has to fit in a box that is approx. 1 foot x 2 foot x 1 foot. It can be a challenge to make everything fit.
As far as freezing the computer. It’s best not to do it. The screen will be the problem. Invest in a good padded carry on case and keep it on your lap on the plane. Sometimes they will collect your carry on bags and stack them in the back, but you should be able to hold on to the computer.