For the benefit of others as this is a public board.

Do not pay the damage out of pocket and then submit. Rather, contact the insurer (in this case through CC) and provide the details. While the exact process will differ around the edges, "Master Card" would have dealt with National and the other driver's insurance and you would have been out of it (other than a release at the end). While you are on the hook in the event that your claim is denied and the other driver lacks funds or his insurance will not pay, that is a rare occurence.

In the meantime, you need to obtain liability insurance for yourself. Lucky that this was only a damage claim and a smallish one at that. If someone had been injured and you had been at fault or the other guy had no coverage, you might have been on the hook for orders of magnitude more.