I think that you need to file your own customer service complaint, seeking the $2,300. Unless you saw what the agent helping you sent on, you have no idea what the agent handling your "gesture" saw.

I would make the request very short and to the point and leave out everything not relating directly to the $2,300, e.g. all of the time you spent and the back-and-forth you had to go through. Make it plain that this is not about compensatory damages (AA will not pay those) but rather about fixing a mistake it has admitted it made. Send a copy of the receipt for the extra $2,300 in tickets. If the whole thing is more than 3-4 sentences, you can forget it. Remember that AA will have an entire writeup of the incident.

Make sure that you compliment the agent who did help you.