I don't know about an enclosed lid, but you can hire Enclosed trailers from many places,
varying sizes and usually used for moving furniture, but can put anything in them.
Before we got out camper trailer, we used a 6 X 4 trailer with a 2ft cage and just put a tarp over it.
These type of trailers are pretty common at Petrol Stations where they hire trailers.
Some national parks have long waiting lists for camping reservations.
When you have to wait a year to sleep next to a tree, something is wrong. ~George Carlin