For at least 30 years (my touring history with flights) airlines had accepted and provided "Amtrak" sized boxes (or close to that size) of up to 126 linear inches that fit the entire bike with both wheels on, bars turned and pedals off.
AA still does accept the large box. Heck, Air Canada used to sell you a big plastic bag and say "good enuf"! It is only in the past ten years or so that the box size has been reduced to 115" by many airlines. I understand why the OP wants to make this work bringing his own big box.
Nothing was more convenient (for me) than using the big box provided by the carrier. I'd pedal into an airport through the automatic doors and then use the bike with panniers still on as my luggage cart. I'd then check the bags, do a ten minute bike pack up and be on the way. Same thing upon arrival.
To re-pack a bike as it was received at the bike shop would be absurd. The asssembled bike is protected from damage by its own installed wheels and inflated tires. No loose parts to keep track of other than the pedals and they can be installed on the back side of the cranks.