Can a Corporation "DBA" as another Corporation?

Yes. I don't know about other states but in Indiana the corporation would need to file an assumed business name with the Secretary of State. We would require the assumed business name before opening the account.

In my state the Secretary of State will NOT allow an Assumed Business Name to include LLC, INC, CORP in the name.

As implied, just because the state would allow it that does not mean that your bank should; i.e. there is only one real entity here and allowing "Inc." in a DBA name is inherently misleading.

Sometimes, corporate ownership is complex and people get the idea that when one corporation owns another there is some sort of a "DBA" relationship. There is not. Each is a different entity.

And it's also true that the folks at the Secretary of State's office are human, and sometimes make errors. The fact that your state's laws don't allow an assumed trade name to include "Inc." or "LLC" or otherwise imply "entity-ship" doesn't mean that the state has never accepted a registration in violation of its own standards.

As Elwood suggested, your bank should not style an account in such a manner.

Agreed with all of the above. As an attorney, the stuff people bring through our doors to open accounts and explain their business structure is tough sometimes. It causes my front-line employees confusion. In a lot of cases, the customer insists an account titled a particular way. If the customer puts up a hassle, we do it for them, but from a BSA perspective, I will treat the customer how they would legally be treated, not the way they want the account setup.

I would just follow the paperwork. Here is New York, when you have XYZ Holdings DBA Apartment Complex, there will be a SoS receipt for assumed name Apartment Complex. If they are XYZ Holdings Inc DBA Apartment Complex LLC just follow the paperwork to understand how the customer has been created. If the customer appears on the SoS website as Apartment Complex LLC....... the XYZ Holdings Inc, is irrelevant, the LLC formed as a separate entity, the customer is just throwing DBA terminology at you not knowing what it completely means.

In many cases, the customer doesn't understand the entity structure the customer paid an attorney big bucks to create for them.