If you are using the b&w case for the Brompton, expect it to get trashed fairly quickly. None of the folders are anywhere near as good as a full-size bike. I have carried on my Brompton using the slip cover that folds up under the seat, but only on flights where I know there is no overhead bin. Other than the checked-luggage size pelican style cases for them, I do not think there is a way either of the bikes will survive being gate checked.

vs the Ticket: ticket packs lighter, if you have the carry the Brompton for long the 25 lbs gets old real fast. The Brompton is faster at folding and can roll on the rack wheels while folded. Brompton front luggage option is pretty super. Integrated pump means you have one less bit of clutter with your other stuff.

vs S&S bike: much longer to put together, but you have a real bike, that's possibly lighter than the folders, and if you have to put it into checked size luggage anyways....S&S case is 26x26x10 - which is roughly the same size as a Brompton case.

vs other options: rentals, raphia clubs, bike shares, or even stashing a few cheapish, but good bikes at repeat locations means a whole lot less to lug around and be in everyone's way.

For all of them: don't forget you probably want to bring helmet, tools, tubes/patch kit,

I find myself taking swim goggles and running shoes more than my Brompton - and I hate running. But the Brompton is really good for public transit and getting the "last mile" around my domicile.