If it is permitted on that card; transfer enough funds onto the card to be able to cover the cost of the flights. e.g. with a $1000 limit : if flights will costs $1050 each , so a total of $3150 , transfer $2200 onto the card beforehand which may allow you to make the purchase

Phone the card company 1st since this will not work on all cards; regardless of a credit balance ,some cards will still not allow a purchase for more than the credit limit amount on the card ( i.e $1000 in your case )

If the above cannot be used, then may be able to split payment over 2 cards over the phone with AA

Failing that, try booking with a travel agent that will allow splitting across cards