I wouldn't worry. You presented a valid form of payment. (AMEX) AMEX refused to pay the charges because they believed your case over their merchant, as any judge would based on the evidence you presented in this forum. They probably told Budget that submitting an unsubstantiated claim may be fraud and Budget backed off. The case is now between Budget and Amex. Have you contacted Budget's corporate office? May be worth a try to start another paper trail.
I had a problem with Budget and refuse to rent from them and don't offer them to my employees very often. I rented a car and a couple days after returning it my wife took a call from Budget demanding to know our insurance carrier. She was caught off guard and told them and took the number of the guy from Budget.
I called. He claimed that I caused an accident in a Budget vehicle. He did not have the details or the police report yet, but claimed it was traced to me. He said the car I "hit" had damage and they were going to bill my carrier. I contacted my insurance carrier and they had opened a case! They were also waiting for the police report to investigate it. I informed both of them that this was not true and I demanded to see the police report.
As it turned out, when the police report came, I was not involved. A misplaced digit in the plate number traced it to me. Budget told me not to worry anymore. (Not even an apology!!!) I suggested they wait until they have their facts before starting the claims process. My insurance carrier (rhymes with Gecko!) told me this happens with Budget a lot and were sorry I had to go through it. "Gecko" got a thumbs up on how they handled the process. I had a direct line to the person handling the "claim" and she didn't seem worried at all about it from the beginning once I told her my story.
Result...no more Budget for me or my employees because of their aggressive tactics!!! The one problem we ever had with Hertz (vandalized window) went very smoothly when they found out we rented with AMEX and they covered it.