The bank is incorrect. Only one of the entities should use The Partnership dba.
If you want to have three entities, then you should. And it looks like you have the income sharing structure correct. And the LLC form will allow you to use the business as a C corp, S corp, partnership or sole proprietorship for tax purposes. You have the flexibility to take the right advice and run with it.
Using The Partnership name for a business implies it is a partnership, which you likely don't want in the event you are sued. Calling yourself a partnership implies that the business is indeed a partnership, and with it unlimited tort liability for both the business and its owners. You should reconsider this name.
I suggest revoking the dba and selecting another name you like. It might seem small now, but a lawsuit that names you individually without entity protection could be expensive to fight off. Much more than the value for use of a name.
I would also caution you to seek professional guidance in your business dealings. Banks are banks, not legal or financial professionals, and their suggestion of how you should structure your business affairs goes beyond their core competencies.
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