No central authority chooses who can join, which is an aspect that I'm sure a bank would love for their private internal database (this is sarcasm by the way)...

Anyone can join the blockchain, not anyone can join the Bank. I'm assume you are being deliberately obtuse rather than the alternative.

Public/private key cryptography is a tool used for many different purposes and by itself has nothing to do with access.

Yes, but combined with blockchain it is used for access.

proof of stake, doesn't make sense in the context of a completely internal database.

Yes it does. You can use a public blockchain protocol for completely internal data, then interface at a later time with that public protocol.

Who is staking,

The validators

what are they staking,

Usually the network token

and how does that prevent fraudulent transactions in a bank's internal database?

Because internal transactions are validated and can be confirmed by external stakeholders (investors, depositors, regulators etc.) on a real time basis. No more Lehman Brothers.

I looked at the energyweb site and it's marketing fluff.

It's real https://explorer.energyweb.org/

When it comes to tracking things on blockchain, once you cross the physical-digital barrier, you end up having to use trust.

Energy Certificates are about transparency more than trust.

So you want to verify that the electricity you got is renewable?

More that you want to prove to everyone that you are buying renewable.

blockchain will only verify that you made a transaction

IoT devices provide trust in the data. TSOs confirm the official values at a later date.

I can send electricity your way which was generated by a gas generator, but how could you tell this from the blockchain?

Because the IoT device reporting to the blockchain is built into a solar panel, not a gad generator.

Also neighboring sources of energy will display similar data. Anyone running a gas generator to replicate a solar profile will be losing money hand over fist.

The answer is that you can't,

The answer is you can.