This is how screwed up Xbox One billing is:
So I relented and got an American Express Bluebird prepaid card for "buying" the legacy Xbox 360 items that are free on the Xbox One, like Games With Gold and avatar promos. (FYI: Bluebird's probably the best deal of all those cards. Very few of the hidden fees most of them have.) I entered my info on the Xbox One. Went to buy the Windows 10 Ninjacat prop. Still got the change-your-payment-option message. WTF? So I deleted it. Went to the official Microsoft Account web page. Entered my info there. Still got the message when I went to buy free 360 stuff on my Xbox One. WTF? WTF? Dug through their support pages. It's a known issue. You have to enter your credit card info on an Xbox 360 console to enable free Xbox 360 purchases on an Xbox One. Deleted my info, again. Luckily, even though I had sold my 360 to buy the One a friend gave me his old 360 when he moved across the country a few months ago. Got it out, connected it, and entered my info. Back on my Xbox One, went to buy free 360 things, viola! That worked.
So if anyone is having trouble buying free 360 stuff on an Xbox One, they have to somehow get on actual 360 console and enter their credit card information there to enable free 360 purchases on the Xbox One. Hope that helps anyone else having that problem.