This question has been discussed a few times.

Unless you absolutely want to ride with a trailer — although it's harder and less convenient than carrying all your stuff on the bike —, the cheapest and most practical solution is to stop at any Brompton dealership to get the original box for free; If none is available, stop by any bike store and perform surgery on a full-size bike box with scissors + duct tape to fit the Brompton. Then, you can chunk it at the aiport with no hard feelings. Repeat at your next flight.

If you do need a trailer, consider getting a Burley Travoy instead.