You did not say which Bike Friday you have, but I assume that you have a model that is capable of touring.
I have considered a tour on my folding bike (not a Bike Friday), but have not done so. But I have three touring bikes to choose from, so the folding bike would have tough competition.
I have flown with my folding bike in a S&S Backpack case. And I have flown with my S&S coupled touring bike in the same S&S Backpack case.
I have never flown with one of my regular (non-coupled, non-folding) bikes. The hassle of dealing with a full size bike in a bike box to get to the airport, home from the airport, in the airport, and unknown ground transportation in another country would make it a no brainer, they would be the wrong bikes for that kind of trip.
Flying international and touring, if the trip was all going to be on pavement, not gravel, I would take the Bike Friday. (I am assuming you do not have a coupled full size bike, if you did you likely would have mentioned it.)
My first ACA tour was Glacier Waterton Loop. Three people had Bike Fridays, two pulled trailers and one had a rear rack and rear panniers. I do have one suggestion, if you use the trailer option, use a flag pole with a very visible flag on it. The Bike Fridays without panniers that were pulling a really low trailer were not very visible from behind. Flag poles with visible waving flags helped a lot.
That Glacier trip, the Bike Fridays with trailers were parked on the left side in the photo.
