New Zealand Law
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You will find that the bank has the power to require all of the sale proceeds to be be paid and not as you have asked. This is in their standard terms and conditions. They can also under the same terms, require that you share your financial information with them. Banks have considerable powers under their documents, and your only remedy is refinancing with another friendlier bank if that is possible. Otherwise if you think what they have done is unfair, you can complain to the Banking Ombudsman
The small print says that they can demand repayment whenever they consider their security is at risk, and therefore when you fell into arrears they must have decided this was the position. Dealing with a bank can be a very one sided process. So they can insist and your only remedy is to try to negotiate
You should first ask the new bank if they will do this, and then you could pay out the old bank and perhaps get the cash you need from the new bank