One suggestion would be to try and make a reservation every week (for 1 week) at the location you returned the car to. Do this until they bill you and take screenshots of the vehicles available. Avis is notorious for hitting people with "loss of use" fees. However, if you can prove they were willing to rent a car every week at that location, you have proof they didn't actually lose money by being sold out and down one car because yours was in the shop.

In my one time being billed for an accident from Avis: They sent me a bill within approximately a month. They hit me for the $50 admin fee or whatever it was, plus the loss of use, plus the damages. The damage was surprisingly lower than I expected, and I sent it to AmEx, along with my proof of vehicles available at the site while "mine" was in the shop. AmEx handled it, paid the bill (minus loss of use, and the admin fee) and I never heard back from Avis. However, it was a bill due to Avis, not a credit card charge.