I'm keeping the stock tyres I have now on my trailer. I do /have gone some places where people look and wonder how I got the trailer in there to start with. I can up size to my truck tyre but I would still carry two spares. I air down the trailer, just as I do the truck (the 75% method) (unless I run into some weird terrain).
My trailer does go airborne from time to time, it weighs 950 lbs, and the tracking stance is different, but that is the way it goes.
The one thing I would get on any trailer is an articulating hitch. A ball just doesn't cut it in the backcountry, it can't take the twisting that it needs to do.