How do I add carrier billing as payment method?

You need to have a supported carrier for your country/region

Availability of mobile phone billing as an Apple ID payment method - Apple Support

You should then be able to select it when adding payment method as outlined here:

Change, add, or remove Apple ID payment methods - Apple Support

Good luck!

Check this Apple article: Availability of mobile phone billing as an Apple ID payment method

https://support.apple.com/en-md/HT205102

That is not supported in all countries.