Use your AA or SPG card to purchase a money order for the amount of your payment. Then you send it in as payment.
And where is it that you're buying your money orders
such that they post as purchases?
Even when it's possible to buy money order with credit cards,
it usually posts as cash advances, which not only doesn't earn you miles, but starts charging interest from day one. Which, in turn, to me is even worse than not being able to buy with a card at all!