I supervise our Traffic Unit and this is why I don't allow simple Driver's Exchange of Info for crashes unless very minor in nature and/or the parties insist, especially when there are conflicting stories.
We would have written the crash report, as stated by both parties, and let the insurance folks handle the negotiations. We would also note that conflicting information obtained during the crash investigation resulted in the inability to clearly establish fault thus no enforcement action(s) was taken (citation).
The way people sue now days, yes, I'd report it to my insurance and give them a heads up. Depending on your deductible, or your desires, you may want to just fix your car out of pocket.