"Billing problem with a previous purchase" notification on iPhone

My wife is getting this error message on her iPhone and Apple Watch... "There is a billing problem with a family member's previous purchase. Ask the family organizer to update their billing information." I'm the family organizer and I have family sharing turned on so her App Store and Music purchases get charged to my payment info. I recently replaced my debit card due to potential fraud, and entered the new debit card number in Apple Pay. It was shortly after this that my wife started getting the message. On her phone, if I go to Settings > Apple ID > iTunes & App Store > Apple ID > View Apple ID > Purchase History, there's another message at the top of the screen... "Your payment method can't be authorized. Update your payment information." When I follow the link, I'm taken to a Manage Payments screen where I see my debit card and an indicator that it's connected to Apple Pay. I see no way to make any updates on that screen. I've made Apple Pay purchases on my phone with the new debit card and it's been just fine, so I know the new debit card info is correct. How can we stop my wife from getting this message?

[Re-Titled by Moderator]