Andrew, I think all you have to do is tell Budget Rent a Car that you want to see the rental agreement for the car that you were driving. Since the Chrysler dealer gave you the car to drive, I would assume their name, or someone who works for them would have had to sign the original contract just to get the car from Budget in the first place.
Since you never signed anything, nor were you asked to sign anything, then the person whose name that's on the original contract that Budget has (in their possession) is the person who is legally responsible for any and all damages to that car.
Just because some "low life" (who was responsible for the accident) doesn't have insurance for them to collect off of, they decide to go after the next person in the chain of events, you.
Don't pay any attention to their letter. It sounds just like the usual "scare tactics" big corporations use to intimidate the little guy.
I hope you don't need to get your lawyer involved. Why should you have to pay anything out of your pocket for something that wasn't your fault?

Good Luck.