For me the question is incomplete to be replied properly.
Why should banks use bitcoin?
It is not only banks. In many branches, industry as example, it is already known that blockchain technology is there, it is known that they will adapt and use it. However, industry still can't find proper business models.
Some speak about different implementation and replacement for proof of work. All of them have their reasons, pros and contras. Banks could call their internal system to whatever they want, even to Proof-of-bank which means techincally nothing, definition could be simply a check by a person working for a bank. All that has nothing to do with bitcoin itself.
I would rather suggest to change the question into: "Why and how should a bank use blockchain technology". As for that question, there are not only answers but real life cases. G&S and other banks did work with Ripplecoin.
With that, I am finding it hard to understand what is here actually been asked. Is it in general why should banks use bitcoin and how should they adapt their systems, then you could look at that similar like a blockexplorer.
If that helps you:
2 years ago I did have that approach to implement blockchain technology in some commercial ERP system, it was SAP. Speaking about banks, your requirements would be probably better covered in Oracle world, as far as I know, it is more spreaded across banks worldwide than SAP.
I did try to write all classes and everything bitcoin related directly in ABAP, for the same purpose that you seem to be speaking about. I wanted to use quite more advanced database like HANA which do fit pretty well to blockchain technology.
So yes, every bank and everybody else can use blockchain technology. You could even store your data. It would contradict some basic blockchain philosophy as if any company does store any data in blockchain, there must be some deletion/cleanup policy, as example for bills which would be required by law to be deleted from the DB after some period of time. In case of blockchain it is a major philosophy problem where I would assume that for file storage and other tasks, there are many good solutions which mostly fit much better to companies requirements.
Why should banks use bitcoin: Banks already use bitcoin. Look at Fidor Bank as example (first it was a Blog, turned to a bank). It is out of question what some do or not, it is free market. I would say that most banks who did research agree, it could strengthen their market position and gain additoinal value
How should a bank use bitcoin: If I would be your consultant, I would strongly suggest to use original sources of bitcoin as banks can't provide such amount of development progress of open source projects. But you should have a team to be able to review and stay in contact with original developers.
Can you use other technologies: Yes, many companies do find Hyperledger good for such goals, because hyperledger is designed to enable multichain interaction. For internal usage, I think it is most documented available solution which I would suggest you to read about is hyperledger.