Let us put it this way... If you were to get pulled over for speeding during the rental and the cop asked to see your license and registration, what would you have handed to him or her? Typically you'd need some sort of paperwork (i.e. the rental agreement) to show that you were the lawful renter of the vehicle. What would you have done at that point?

When they handed you the keys to the car, did they give you any sort of paperwork with it? They should have unless you were a loyalty member, in which case your rental agreement should be accessible via rental company's mobile app. This is standard stuff. If you didn't get it, or you misplaced it or threw it out, then that's on you. If the claims adjuster demands it, you're going to have to contact Avis to get a copy or they are not going to process your claim. Not sure what else to tell you.