You don't have to buy a new phone, but you will have to borrow one to set this up (and potentially make any transactions once it's set up). There is almost no way I would recommend trying to fake out the registration process as you would likely get snagged for anti fraud steps they have in place to protect identity.
I love you are rocking an older phone and don't want to upgrade when it so far has worked for everything you need. Reusing a device is better than recycling and not replacing is even better for everyone. Also, who knows if Apple is rolling it out slowly and once the bank and systems are working with the "newest" they don't push an update or a button to roll it out more widely?