First, I would advise to review the format of the field used on both sides, as custom lists can be saved as text strings, or as a numeric fields in some cases (meaning the field itself is just the number, and the label for the text of each option is saved separately).
Not sure if this is the same documentation you made reference to, but the process for mapping custom fields is detailed here: https://docs.exalate.com/docs/how-to-synchronize-custom-fields-in-jira-cloud If you have already followed the process and confirmed the format of the fields used for these radio buttons, could you provide said formats and a snippet of the incoming and outgoing script you are using for the field in each side?