"I prefer that they are so puncture resistant as possible, but at the same time roll as easy as possible."
Those are basically contrary characteristics as you probably know. On the plus side, these Kevlar belted tires they have roll great and are really puncture resistant compared to anything we used to have. But once you insert 4 mm of plastic to keep the thorns at bay, it's no longer a fun tire.
The tire you run is based on the surfaces you run on not the frame you have. For that trip 32mm is fine, though I prefer 37 myself, or run them anyway. I've only done about 2000 miles of that trip, but met lots of folks who were on the trip, and running 32mm tires. My neighbor has done two cross country runs with support, they basically race it, on sew-ups. the roads are basically the same roads locals train on with their road bikes except for a transcan you are running through the whole lot of them.
I would add to the mix a road morph pump, or possibly the rip-off MEC is selling, a quick stick, and some patches including PSA ones. Choose a tire that is easy to remove. With my tire they hardly even need the quick stick they just fall off when I need to change them. If you look at the White site above you will see mention that the armoured tires are hard to mount.
So do you want armoured tires that roll like crap, and never get a puncture, or tires that might get one or two puncture in your whole transcan, but roll like magic, and are easy to fix. For me there is no comparison. The fat tires will be like rolling through sand. I have fat tires on my comuter, because every 10 years or so I get sucked into a trolly track, but with the fatsos it's no problem. But I don't particularly like the way they run. In Toronto, the urban roads are mostly worse than what I have covered on the Transcan.