With the risk of sounding redundant... other suggest Artic Fox, which is one of the few actual 4 season trailers. It is small and heavy for its size; undoubted too heavy for your SUV.
I wonder if the campground you have in mind is open in the winter.
If this is a once or twice in decade event, why not rent... or better yet, buy a nice used motorhome that can "sleep" your family... which is a much easier and comfortable way to experience this great country; especially for stop, see, and go traveling, then sell it when you get back. Of course, a MH could be excellent in cold weather abode.
To your 3rd question, you might have to winterize between visits, otherwise heat has to be kept on, including basement. Although water lines could be disconnected and taken in and out to prevent freezing.
If one could be bought right, it could be less expensive than renting.
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