Snow Leopard to El Capitan upgrade

On the old Mac have you tried entering your ID and at the end of that enter the code send to one of the newer devices that support two factor natively?

What if I use two-factor authentication on a device running older software?

If you use two-factor authentication with devices running older OS versions—like an Apple TV (2nd or 3rd generation)—you might be asked to add your six-digit verification code to the end of your password when signing in. Get your verification code from a trusted device running iOS 9 and later or OS X El Capitan and later, or have it sent to your trusted phone number. Then type your password followed by the six-digit verification code directly into the password field.

Above from:

Two-factor authentication for Apple ID - Apple Support

If your MacBook will run El Capitan and is new enough to have two factor authorization then you can download that to the MacBook and transfer it to the Minis. You can get El Capitan here

https://itunes.apple.com/app/os-x-el-capitan/id1147835434?ls=1&mt=12

Also, regarding the payment method message, when is the last time you "purchased" something from the Mac App Store with that Apple ID? If not recently , "purchase" something and then try again on the Minis.