If the app uses iCloud to backup/use its data and the 10.2 version of the app can read the data from the 8.2 version (you can contact the app developer to verify), then you should be able to just download the banking app and use the old data.
For example, I've used many apps on my iPhone's on versions all the way back to iOS6. With each new iPhone I setup as 'new', the apps that are designed to use the iCloud data simply work. Of course, I have to make sure I perform a backup of that specific app's data to iCloud before I upgrade, and you'll want to do that too.
I'm very happy with how many well-written apps are using iCloud. I know that many other people don't have the same experience. It looks like you're able to use both iPhones at the same time so you should be able to test and make sure your data looks good on the new one before making the permanent switch.