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I have one of these that you're welcome to borrow https://www.vincita.co.th/collections/folding-bike/products/b132b-soft-transport-bag-for-brompton-bike
It's OK, not particularly padded so you may want to look at padding out with bubble wrap and clothes (it's big enough that you can pack quite a bit around it, I used this plus hand luggage for a two week work trip to NY) but it does the job.
Oof. Doesn't look good!
Can I ask a favour, does anyone in London have a Brompton hard/soft case I can borrow for a long weekend in June?