unable to replace obsolete payment method

I have a single credit card registered with apple. The card was lost so I replaced it. Now, I want to update the payment method. However, if I try to "add payment", a popup appears that says "one or more payment methods requires verification." On the other hand, if I try to "remove" my old credit card data, a popup appears saying "Unable to Remove: family organizer requires one or more payment method." Finally, if I click on verify, the security code for the old card is required, which is something I don't know.

I seem to be trapped within these three dead ends. Does anyone know the way out?