I was just looking at my National Rental Car agreement last night. I'm not going to be able to quote precisely, but the language in the National agreement relevant to your question basically says that you agree to pay for loss-of-use whether they would have been able to rent the car or not.

Depending upon how much you've rented from Budget in the past, and how much you plan to rent from Budget in the future, I'd consider trying to negotiate something with them (including the elimination of the $50 "processing fee") or just ignoring it. They shouldn't be out any money from damage you caused to the car (if you did, indeed, cause it); but it shouldn't be a profit center for them either.