In theory you can use the shopping portal to buy prepaid visa cards (using your AA CC) and then use those cards to buy the airfare. You get 3 miles per dollar on the shopping portal and 1 mile from the CC that you use to buy the prepaid visas. so that's 4 miles per dollar spent, and all count as loyalty points.

Your biggest problem is going to be that the cards have a $250 max ($500 if you get the business ones, but they have fees and shipping charges) and so you will have to find a way to pay using multiple cards. As far as I know that's not possible on the AA website. You could buy tickets at the airport but I'm guessing that you will have trouble using virtual gift cards there (the cards can be added to Google pay, but that requires a point of sale that accepts tap to pay, and a willingness to accept payment split over multiple cards). It's possible that a phone agent will allow payment split over multiple cards, though I think there are still phone booking fees?

Another option is to use the prepaid visa cards to buy AA gift cards and then buy the travel using those. There's still a limit of 8 gift cards per ticket purchase; I'm not sure if this will lead to issues for you (does the 4.5k worth of travel need to be on one PNR?)

I'm not necessarily suggesting that this is a good idea by the way; I'm just pointing out that it's one way to use the portal to get extra miles for your AA ticket purchase. Personally I put all my airfare on the Amex Plat.