Well, I did trap 2 wild rabbits years ago, before I started with bunnies, but even being european rabbits they would always be stressed out in a cage and would always try to escape, with no regard to their safty and health. They are wild animals, unable to grasp the concept of a cage being their whole world.
I had a litter sired by a wild rabbit (caught that guy and set him free 10km away), even the offspring was different. Coping with close quarters and being not scared of humans is a for centuries bred in trait of our domestic rabbits, Sold one of these does to a farmer who wanted to repopulate his farm with wild rabbits but had already two wild rabbits (bred in captivity by an expert) die on him, so he was happy to try the half wild doe.
I'm pretty sure Cottontails are much less suitable as lifestock or pets, like hares they do not live in burrows, so there most likly is very little tolerance for close quarters. And even kits that seem to be docile will get skittish and wild when they grow up - as far as I know Cottontails don't live in groups or colonies, so no genetic disposition for friendly social behaviour.
Anyway, they are pretty much useless for meat, you can't show them, they will never be pets, and so on.
Better try to find a way to get rabbits of a breed that suits your plans transported to you, isn't there some kind of platform where private people offer to transport stuff on trips they do anyway for a reasonable fee?