I've finally gotten things straight with Avis. After months of being ignored by my claims officer, I got her boss's number and gave him a ring. He told me that Avis had a policy that if a renter is at fault for an accident and they total out the car, Avis & Budget (Cendant) will no longer rent to them. He then paused, and admitted that he didn't know why I was dropped, since the deer was clearly at fault in my accident. We both agreed that Cendant, Inc. and its subsidiaries should no longer rent to that deer.
So that was it. It wasn't Avis being stupid with me. It was one individual within their office being stupid with me. I guess every office has one of those people.