Purchase Xbox 360 content on Xbox.com without credit card on file
So there's finally official confirmation that the real answer is "You can't." From this Major Nelson post about Xbox 360/Xbox One backwards compatibility:
The Xbox 360 Marketplace and the Xbox One Store websites use different architecture, so if you have credit but no payment option associated with your account, you won’t be able to use the credit to purchase Xbox 360 titles. You will have to add a payment option or purchase the title using an Xbox 360 console.