If you had tried to book this on-line, you wouldn't have been able to. If the counter/office is closed,it's closed.
It might have been prudent to directly contact the local office, not just a CS agent.
To expect an employee to stay after working hours for someone, who may or may not show, on a weekend is to have much faith in their fellow man.
I'll be surprised if Avis does much for you. Your best argument (there will be no tape) is a copy of when your plane arrived and the "logic" of the situation.
By thw way, don't you receive email confirmation on your Avis reservations?