The seat numbers suggest that you were flying first class. The air personnel is much more forgiving when you are in that category. I flew few times with a Brompton as a carry-on and found it far too much stress than it is worth. Even though I was successful I expect statistically to end up with charges for a bike or excess luggage and haggles with air personnel. Putting the bike into check-in luggage involves no stress if you meet all limits.
Not First Class.
1. British Air was coach plus.
2. Virgin was coach.
3. Southwest only has one fare.
One theory, and it may be a bit crazy, but since I am over 6' 2" and not slight, the bag may not appear so big in comparison.
I'd say out of all the airlines, Southwest was the easiest.
Also, at Heathrow security, I recommend letting them know that it is a British Made Brompton.
Security at San Francisco and most US Airports know about these bikes. I recommend getting onto TSA Pre-Check to make things easier.
Removing the seat/post (the telescoping one) makes it easy for the bike to get thru the scanners.
The packed bike:
FYI: Being able to do this was a MAJOR reason I bought a Brompton. I was doing a lot of business travel and I decided I wanted to have my own bike with me to avoid taxi/Uber fares. I also got pretty good at packing, being able to do a week or more with everything in a T or C bag, including laptop. In 2015/6 I probably did over a dozen flights with the Brompton (I was bi-costal for a time).
Now, less flying (thankfully) but with Amtrak's new bike policy the Brompton is the bike I have as my sole transportation during the week.