As you are not a Medicare provider at all (or so as I understand), I have a strong doubt if you would even be able to submit a claim electronically as it might get rejected stating "Invalid provider primary identifier".
As the patient was a walk-in and was informed by your office that you don't treat Medicare patients, you may get an ABN signed by the patient, just to be on a safe zone, so that even you find the payers' doors shut, you have the other party to collect the money from.
As many older patients misinterpret (or misunderstand) information given by payers, I would suggest you to call the 2ndary payer yourself to understand how to submit a claim in this situation.
I am curious to know how this might turn out. Please update us when you find a solution. Good luck.