If the supervisor had made any statements or claims to counter your claim make sure to follow-up on them. This may involve getting a second opinion from a similar business. This will let you know if you have a leg to stand on. If you get complimentary information you can still proceed, but contradictory information will give you all the fuel you need. Put this in your arsenal to use with the manager or general manager.
Confidence when speaking to the manager can be a decision-maker.
Tell them, politely, what they are going to do for you, don t ask them to do it for you. Politely and simply list your points without the long drawn out story. If they ask questions or make points, counter them and mention any discrepancies you have learned via the supervisor or your second-opinion. If not, refer above to the two agencies that you can use to escalate your case. It is a chain of decision-making power.