First off, there is absolutely an initial rental agreement. What you get at the end is called a "receipt". If Avis told you otherwise, then their representative misspoke.

As far as the card used when the vehicle was checked out vs. the one the final charges were billed to, it's fairly easy for the credit card company to see what's what since Avis would have run a pre-authorization (a.k.a. a "hold") on the original card at the time contract was initiated. You can change payment methods at the end of the rental, which is what it sounds like you did here.

Will they check? Maybe. Will they deny your claim on the basis that the card you're trying to file a claim against was not the one that the pre-auth was run on? Maybe. We've seen folks change to a different card at the end of the rental and their claim was honored. It probably depends on how much the claim is for, how closely the claim manager wants to dig, what card company/insurer it is, etc.

The bottom line here is that it sounds like the train has already left the station and you are going to take a little trip. Whether your train arrives at the final station safely or it crashes along the way is hard to know. It is what it is at this point. There probably isn't much you can do other than to just collect the documentation they are asking for and hope for the best.