The strict anti-money laundering requirements imposed by the authorities on the banks means that the answer to your question is probably 'no', but if they really wanted your business they could always visit the home to satisfy themselves that there was another person.
That said, many bank accounts are opened online now with no need for a face-to-face application, so why not try that route if your high street bank is awkward?
A Bank would usually want to investigate the use of a Power of Attorney, since if you are a new customer, they cannot be sure that it has been properly granted unless it has been registered at the Family Division of the High Court.