Can I be found at fault if they file a claim with my insurance company?

Under circumstances you have described, yes, you can be found liable for this incident. Though the amount you stated might seem a lot to you, some very minor accidents result in somewhat large repair bills, these days. You can ask for a copy of the repair estimate and take it to another car repair shop and ask them if it seems reasonable to them (showing them some detailed photos of the damage to the car will definitely help, and might be necessary). If your car repair place tells you the estimate is high, you can argue it with the other driver, but you cannot force him/her to accept a lower amount. You will likely have to decide whether it is worth paying the damages and avoiding an insurance claim or letting your insurance company handle things for you. BEWARE, though, if you pay the requested amount, unless you get a signed release of liability from the other driver, you have no guaranty that the other driver won't sue you anyway and/or make a claim against your insurance policy for bodily injury (and do not ever assume that just because the accident was minor the other driver will not claim to be injured...even if he/she really wasn't).