my brother was able to travel [just 2 months ago] on a domestic flight with a brompton and he put it on the overhead bin just above his seat
no cover no bags no straps just the seat was removed and I think he removed the drive side pedal.
this was on a Southwest Airlines from Los Angeles to Texas im guessing he flew on a Boeing 737 didn't even know the overhead bins where that big to fit a brompton.
he was asked a couple of times prior to boarding and when boarding what it was and politely saying its a mobility assisting device/apparatus
there was no damage at all to the brompton
the brompton is painted ALL BLACK[like most wheelchairs] it does look like a tiny weird looking wheelchair when folded to someone that is not familiar. if he left the brooks leather seat in the bike i think no one would have believed him.
the brompton bicycle folded should just always go where the baby strollers or mobility devices ie folded wheelchairs, walkers go. there is a specific area where these are kept inside the cabin until passengers with babies, old folks, persons with disabilities disembark to claim them. basically should be allowed checked-in as a stroller.
Reality is most airline personnel are unfamiliar with their own rules and the word "bike" "bicycle" triggers them to charge fees. im not sure what kind of trouble you could get into these days for not saying so. An alternative answer to the question to "what is that your carrying Sir ? " answer: " oh these are some bicycle parts for my 6 year old sons bike. "why are you hand carrying it ?" answer: I just want to surprise him when i arrive or something to that effect