Thing is, go on, call yourselves (and chiros) "Doctors" and be "right". Put it on all your cards/letters etc.
Then when on a plane or ship, when a Doctor is needed for a medical emergency, and you are then identified, i hope you know what to do. It isnt uncommon for crew to scan the passenger manifests looking for a doctor when need arises.
This isn't a thread about people with PhDs calling themselves 'Dr' (despite the first few replies which didn't' actually address and rather missed to point of the OP)
Someone with a PhD is perfectly entitled to call themselves 'Dr' and book plane tickets using the title - though if they get a free upgrade as a result they prob ought to state they're not medics.
But anyway this is about Chiropractors calling themselves 'Dr' - your plane example is perhpas even more worrying in this situation - whilst I'd be reasonably confident that a lecturer in English literature will swiftly declare that he's not a medic I'd be slightly more worried about the potential response of a Chiropractor in your scenario.