As I understand your question, you really only want one business, the nonprofit, but with the same name you have now. In Washington, where I practice, you'd accomplish that by first changing the name of your existing business and then forming a new company with the same name as the old one. You wouldn't be able to have two companies sharing the same name simultaneously, even if one is a nonprofit and the other is a for profit.
Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.
While you may be able to have the two businesses with the same name, it is an extremely poor idea and a big red flag to the taxing authorities. The ability to convert payments to the for-profit business into tax exempt donations to the non-profit is too big a risk to go unchecked. You'll be asking a for a detailed audit. I had a client that did this and it was not well received. At the same time, your reason for doing this may be valid, but it will be assumed to be calculated to lead to fraud on the taxing authorities and contributors.