I know that Budget and Avis have an affliation. The last vehicle we rented through Budget...and they didn't have a vehicle in our category...they just called over to Avis and they brought it around for us.
It is curious as to why you did not wait for the Avis personnel to document the time of the car's return? This would have been definitive. If the Budget counter had been still open ....would you have processed the return in a normal fashion? You don't mention if perhaps you were running late to catch a plane etc...
I know that whenever we return a vehicle ...I always make sure that time of return is noted on the slip, ...so that the appropriate charges will be assessed.
Did the return box not have a time stamp??(Punch clock type )
So, this is a good warning for others that may be tempted to follow in your steps.