The PoS option was to highlight that power consumption doesn't have to be an issue. Of course, PoS has its own issues.
The network can use any other type of proof, like Proof of Authority where only a buch of validators owned by the banking system can process the transactions. The network can be even tokenless, no profit or incentives from it, just the secure architecture.
The keyword is "private." The redundant system all the banks maintain can be reduced to a private, permissioned blockchain, creating a network for the banking system to handle their own transactions in addition to a seamless inter-bank communication.
I doubt a network for just one bank can be that useful compared to the current situation.
Also, I'd say that every bank has (had?) a team researching the blockchain.