In general, cashing checks payable to a business is a lousy banking practice. It facilitates income tax evasion and, if the business is a separate entity from the owner, may facilitate fraud against the business or its creditors.
That's the foundation point, but your example is slightly different because you are not cashing the checks, you are splitting the deposit or giving cash back. That means there is still a paper trail tied to the business receipts.
The other variant is your reference to a "DBA." For some bankers that means a sole proprietorship. However, any business entity can do business under an assumed name; e.g. corporations, partnerships, LLC's etc. DBA is not a synonym for a sole proprietorship.
Assuming your business is a sole proprietorship, splitting the deposits is probably not a significant issue. If the business is an entity, it's a lousy banking practice.
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.