Think about this, you pay the other driver $1000 to get the car repaired but they do not. They pocket your money and file against you anyway. You will need an unrelated witness to back up your story. File the claim, it was your fault, that's why you bought the policy in the first place.
Read your policy. You likely have a contractual obligation to report the claim. The last thing you need is for the driver of the other vehicle to file claim a month from now alleging cervical injury and wanting $20,000. Not reporting an accident where you are liable is being pennywise and pound foolish. With that amount of damage, many states require reporting to the police as well.
