First of all, I don't believe that premise of your project complies with consensus architecture. As for my knowledge, you'd need to rethink many aspects of blockchains themselves in order to have truly "private ledgers" on selected nodes, and the question is whether that would work. This is centralised way of thinking.
One way of achieving what you want is encrypting the data and then publishing it on Ethereum network. However, the sentence
The transaction must be added only to the ledgers of the transacting nodes.
means that you can't use this approach.
Another one is maybe using the Polkadot project developed by Parity (former Ethcore), which would give you the ability to have many transactions between users, and publish the sum later on main blockchain. But, as you can clearly see, the sum of those transactions would be visible if not encrypted by the bank's public key. It's worth noting that this isn't production-ready at this moment.
There is an initiative called Project Alchemy, which the ZCash and Ethereum developers are working on. It is based on zero-knowledge proof, which (as I understand it) in short means that you could have transactions that are private, secure and also public.
Last time I checked it also wasn't production-ready.
More useful links:
Integrating ZCash and Ethereum (ZoE)
Congratulations on coming to Ethereum and good luck!