Ooo yay! My husband and I both travel with Bromptons. 
We had trouble finding a hard suitcase when we first got them last year, so we ended up going with a soft case instead. We’ve checked them many times now on both short and long haul trips and never had any problems! We do pad out the bike a little bit in some areas, but we’ve talked to the manufacturer of our bags and they said they’ve never really used extra padding.
We use the Sightseer set up from Vincita: https://www.vincita.co.th/collections/bags-for-folding-bike/products/sightseer
The insert that comes with the bag is great too! We put all of our clothes in there so the bike bag is the only checked bag for each of us.
The only thing that’s a bit tricky with this bag is that I find it a bit hard to wheel around (since it only has two back wheels.) We’ve been to their shop in Bangkok though and they’re apparently working on a four wheel version, but I have no idea when that will come out.
Highly recommend anything from Vincita. Super durable bags and they have great customer service!
Oh, and if you haven’t bought your Brompton yet, I highly recommend getting the telescoping seat post, the mounting block on the front, and the rear rack. They’re a bit extra, but we’ve found them to be really useful!
