They still have to pay.
IMO, all they have, even if the authority accept that the car was being used by or on behalf of a bona fide BB holder, is mitigation. Not because they forgot to display the BB which would have permitted them to park for up to 4 hours for free, but because they claim that they've done this before without issue and did not know such a requirement existed in non-designated bays.
All the driver can do is try.
Who is the registered keeper and are their DVLA details current? I ask because it's unlikely that the authority would cancel the PCN and therefore the next stage, the NTO, should be considered. A NTO is sent to the registered keeper.