Accom Supplement is done through WINZ and all you have to provide is income, rates, mortgage, insurance and something else which I can't remember. I wouldn't worry about the bonus and just give them a couple of DH's payslips to show them the income.
As for banks I have heard that Kiwibank in Palmy is really bad for customer service. Hopefully BNZ will come to the party here as I think they are wonderful.