All right, next trip with an M6R completed. Summing up the experience so far, the security appears to be no problem, to be contrasted with gate agents. Again, optimal for plane entry has been the moment of stampede onto the gate. On the first leg, the agent grabbed the side of the Brompton hidden under the cover. We switched to a tale that it was a foldable walker. After all, you can walk with it

. Seemed to work a tad better than a stroller claim. (A handicap sticker could help and the Brompton logo right at the center of the bag is of no help.) We further found that we could fit the Brompton into an overhead bin of a Boeing 757. The fit was tight and worked with wheels directed into the bin, away from you, i.e. the same way you put the Brompton onto the conveyor belt.
Otherwise, it looked that it might be good to equip the Brompton with some luggage handle attached to the frame and sticking out from the inside of the cover. One more observation is that the black color of the Brompton's frame might be optimal for the operation, as minimizing the attention, when the cover needs to be open from top, in order to grab the frame or to attach the Brompton bag.
I am sorry about the AU situation. In my experience, the check-in and gate agents are much more flexible when flying out of the US than when flying in, so I would be hesitant trying out the carry-on option within an international travel. The travel with a NWT in a hardshell suitcase is now for us routine and I've also done international with a Dahon in a soft bag.