You have a very interesting problem that I've looked into for other reasons, but it would involve swapping the Chubby for a Burley Travoy and the DIY cardboard box for checking the Brompton. With the Travoy, large duffle/backpack, and T-bag, I estimate ~100-160L/36-46kg (flying-riding) max gear capacity that would pack into 2 checked and 1 carry-on of standard US airline baggage size/weight, and the typical excess baggage fees for 2 checked items seem to a nominal <$100, and it's 100% legit (ie, no need to claim 'sporting equip' or 'mobility device'). Once you exceed 2 checked/1 carry-on of standard size/weights, the airline fees seem to increase exponentially.

Don't know if you've ever tried a Travoy, but they ride very nicely with weight and mirror the Brompton's fold and versatility. For example, you can easily store everything (bike/trailer/gear) inside a 2 man tent with you, both bike and trailer can be wheeled around indoors like luggage, and if you are in good shape (afterall, it's 50+kg total), in <30sec you can convert from riding to full backpack/hand carry for staircases, public transport, or a short bushwhacking hikes to secluded stealth camping.

If that's an option you'd be interested in, let me know.