Without reading the specifics of the Amex coverage, most insurance in such a situation includes a "right of subrogration" clause. This means that after paying you for the claim, Amex reserves the right to go after whomever was responsible for the accident to recover the money they paid you, and you agree to cooperate in such an effort (such as by providing testimony or any supporting documents). It would almost certainly cost Amex more than $320 to do so, even if they could prevail and collect, so they just let it be.