I would simply get them to put a clear email stating the time they got to the gate and simply put a claim in clearly stating that the compensation is required under EC261
Also state time that they arrived at the gate; highlight the 10 minutes as per CoC and include a copy of boarding pass to show that they were checked in
If it still invents garbage, then time to simply take to a small claims court
I cant remember if AA has a specific email address for EC261 claims