Tender the matter to your auto insurance carrier (that is what you are paying them for), or pay the bill.
If you elect to pay the bill, make sure to get a "settlement and release of claims" signed by the owner prior to tendering the money.
The alternative is to get into a lawsuit with the auto owner over the scope of your negligence/damages in small claims court. This is not a very viable option for the amount of money that is being demanded. You can of course go that route if you want to, but understand that if you do, you will have a court judgment for negligent operation of a motor vehicle entered against you (civil, not criminal) which can potentially affect your auto insurance rates, just as much, if not more than, any claim against your auto insurance.
You can try mediation - use a third party neutral to help you reach a resolution - but again, with the amount of money in question, you are probably going to spend more on the mediator plus the time off work, etc. than you would simply paying them. If you do want to try mediation, contact your local bar association and ask for referrals, they can provide you with a couple of attorneys or professional mediators in your area - you can then contact one or two that you like, and they will help you with setting up a mediation with the other owner.