A colleague of mine a few years ago was executor for an uncle, and the banks were demanding various ID that she didn't have - she had no driving licence or passport and wasn't about to acquire them just for the banks.
In the nicest possible way she could be quite stroppy - she held her ground, maintaining that since there was no,legal requirement for her to hold either pp or DL, they could not demand them.
They gave in in the end - I forget what they did eventually accept, though.
In the OPs circs I think I'd contact their head office/customer services, AND one of those money-problem pages, like in the Sunday Times or Telegraph. As someone else has said, it's probably often a case of staff afraid to use common sense and thus maybe get into trouble when the computer is insisting on boxes being ticked.
But this is not exactly an unknown problem, and banks really should be a bit flexible in such circs. However it's not unknown for bank staff to be pretty clueless at branch lever even about power of attorney and executor-ship - training seemingly sadly lacking all too often.