Instead of buying GC's or using Amazon Payments to make the spending threshold find a friend or family member that's looking to buy an appliance, furniture or a TV and buy it for them on your Amex (obviously they'd pay you by cash or check). Once the charge goes through pay it off right away.

I agree with the other posters that no one here knows what triggers a FR, but a new card with GC or Amazon Payments spending will likely throw up red flags. A FR for you right now would probably end your Amex relationship for now. The difference between "income" and "projected income" is probably a deal breaker for Amex.