First of all this is a very old topic!!!
To answer your question, well you got how you treated your bank.
It doesn't look like your bank refused to give you the references, they have simply asked for more information which is only a standard thing to do.
Like it's a standard practice for banks to ask for a reference letter from other banks before opening an account. It is a standard practice to provide your bank with the necessary information, so they could provide the references you need.
There's no shame in getting back to your bank and providing them with the info they need to get the docs that are required.
Also, in terms of the details for your bank, there's a room for some flexibility, so sometimes there's no need to provide the exact name of the financial institution that needs the doc, but you can talk to your bank and just be transparent with them that you want to open an account in another bank (for whatever reason) and that you need the references for this purpose. You can ask for this letter to be as general as possible rather to address to particular bank and this is a very nice letter to have.