1) Set up an offshore mailbox with mail forwarding. There's one in NZ that comes to $7 a month for example, and another in Malaysia that's $28/yr flat, plus forwarding fees.
2) Set up a USA2Me account (TX maildrop address) but reference your offshore maildrop address. Pay $23 one time fee for the maildrop (use a gift card to pay, no other fees except forwarding mail to the offshore maildrop).
3) You can use either real or faux info for the TX maildrop, your choice. Because you used an offshore address, you don't fill out the US government form to identify yourself.
4) Apply for the card account you can get using the TX info. The card goes to your TX box, then will get forwarded to the offshore mailbox, which in turn will get forwarded to you. With the de-facto anonymous card received, anonymous payments are then possible. The mail forwarding expenses are the cost of attaining privacy and direct control over your anonymity at each step, as opposed to trusting third parties.