BikeFriday makes a kit to convert the hard sided carry case into a trailer. That would work as checked luggage, but it won't come close to qualifying as carryon. It's a great folding bike, but I think the suitcase/trailer would be a nuisance. I would just pack the bike in a box, discard the box at the airport and ride away. When you go home, just put it in another box, andleave that one at your airport.

I've also left my bike in the airline luggage office, gone home and come back with the car to pick up the bike, but that might defeat your purpose.

It's also possible, if your bike packed in a soft bag looks close to fitting in the overhead bin, that you can carry it through security. If it won't fit in the overhead, they will probably gate check it for you, saving you the fee and returning it to you as you leave the plane.