IMO dont 'ask' for the $300 back (assuming it **was** an error/glitch)

Contact your CC company; provide a print out of the receipt, the rules from the AA website, explain you were overcharged. Lets them get the $300 from AA.

If you 'ask' AA you will spend 6 months chasing them. And fail.

Do come back and tell us what the receipt actually said. Easiest is if agent mistakenly charged for each segment...