You could create a 'virtual' anonymous cc card by combining several services, but it's cumbersome. Get a faux profile from fakenamegenerator, a forwarding number from tossable digits, and 2 private mail drop addresses (one offshore, one virtual like USA2Me). Get the offshore one first, then the US one using the other maildrop address when applying (pay for it with a gift card). This way you can get a real US address for a faux name.
Some of the cards in the gift card display area at the store advertise they are 'real' credit cards. Enroll using the fake name and social, but real US mail drop address and forwarding number. The card is mailed to the US maildrop, gets eventually forwarded to you, and you verify the phone using the forwarding number. You now have an anonymous credit card.